Hamilton cycles in vertex-transitive graphs of order a product of two primes
aa r X i v : . [ m a t h . C O ] A ug HAMILTON CYCLES IN VERTEX-TRANSITVEGRAPHS OF ORDER A PRODUCT OF TWO PRIMES
Shaofei Du a, , Klavdija Kutnar b,c, and Dragan Maruˇsiˇc b,c,d, , ∗ a Capital Normal University, School of Mathematical Sciences, Bejing 100048, People’s Republic of China b University of Primorska, UP FAMNIT, Glagoljaˇska 8, 6000 Koper, Slovenia c University of Primorska, UP IAM, Muzejski trg 2, 6000 Koper, Slovenia d IMFM, Jadranska 19, 1000 Ljubljana, Slovenia
Abstract
A step forward is made in a long standing Lov´asz’s problem regarding hamiltonicity ofvertex-transitive graphs by showing that every connected vertex-transitive graph of order aproduct of two primes, other than the Petersen graph, contains a Hamilton cycle. Essentialtools used in the proof range from classical results on existence of Hamilton cycles, such asChv´atal’s theorem and Jackson’s theorem, to certain results on polynomial representations ofquadratic residues at primitive roots in finite fields.
Keywords: vertex-transitive graph, Hamilton cycle, automorphism group, orbital graph,finite field, polynomial.
Math. Subj. Class.: 05C25, 05C45.
The following question asked by Lov´asz [42] in 1970 tying together traversability and symmetry,two seemingly unrelated graph-theoretic concepts, remains unresolved after all these years.
Problem 1.1 [42]
Does every finite connected vertex-transitive graph have a Hamilton path?
No connected vertex-transitive graph without a Hamilton path – a simple path containing allvertices of the graph – is known to exist. Moreover, only four connected vertex-transitive graphson at least three vertices not having a Hamilton cycle – a simple cycle containing all vertices ofthe graph – are known so far: the Petersen graph, the Coxeter graph, and the two graphs obtainedfrom them by replacing each vertex with a triangle [10]. None of these four exceptional graphs is aCayley graph, that is, a vertex-transitive graph admitting a regular subgroup of automorphisms.This has led to a folklore conjecture that every connected Cayley graph possesses a Hamiltoncycle.Problem 1.1, together with its Cayley graph variant, has spurred quite a bit of interest in themathematical community, resulting in a number of papers affirming the existence of Hamiltonpaths and in some cases even Hamilton cycles.Such is the case for instance for connected vertex-transitive graphs of orders kp, k ≤
6, 10 p, p ≥ p j , j ≤ p , where p is a prime [1, 14, 36, 39, 40, 48, 49, 52, 53]. Furthermore, for all ofthese families, except for the graphs of orders 6 p and 10 p (and of course for the Petersen graph, This work is supported in part by the National Natural Science Foundation of China (11671276). This work is supported in part by the Slovenian Research Agency (research program P1-0285 and researchprojects N1-0038, N1-0062, J1-6720, J1-6743, J1-7051, and J1-9110). This work is supported in part by the Slovenian Research Agency (I0-0035, research program P1-0285 andresearch projects N1-0038, N1-0062, J1-6720, and J1-9108), and in part by H2020 Teaming InnoRenew CoE (grantno. 739574).*Corresponding author e-mail: [email protected] p -group has a Hamilton cycle (see [71]), Alspach’s proofthat connected Cayley graphs on dihedral groups of order divisible by 4 are hamiltonian (see [4]),and Ghaderpour and Witte-Morris’s completion of the proof of existence of Hamilton cycles inCayley graphs arising from nilpotent and odd order groups with cyclic commutator subgroups(see [24, 25]). And finally, with a combination of algebraic and topological methods a Hamiltonpath and in some cases even a Hamilton cycle was proved to exist in cubic Cayley graphs arisingfrom (2 , s, pq , where p and q are primes. Theorem 1.2
With the exception of the Petersen graph, a connected vertex-transitive graph oforder pq , where p and q are primes, contains a Hamilton cycle. Since vertex-transitive graphs of prime-square order are necessarily Cayley graphs of abeliangroups and thus hamiltonian [47, 48], the primes p and q may be assumed to be distinct.The proof of Theorem 1.2 depends heavily on the classification of vertex-transitive graphs oforder pq from [58, Theorem 2.1] (see also [54, 55, 56, 57]), based on whether these graphs door do not admit a particular imprimitive subgroup of automorphisms. Three mutually disjointclasses are identified. The first class consists of graphs admitting an imprimitive subgroup ofautomorphisms with blocks of size p (the larger of the two primes). The second class consistsof graphs which admit an imprimitive subgroup of automorphisms with blocks of size q , but nosubgroups of automorphisms with blocks of size p . Finally, the remaining graphs are characterizedby the fact that every transitive subgroup of automorphisms is primitive. The first and the secondof these three classes are exhausted, respectively, by the so-called metacirculants and Fermatgraphs. In short, a metacirculant is a graph with a transitive cyclic or metacyclic subgroup, anda Fermat graph is a particular q -fold cover of a complete graph K p associated with the actionof SL(2 , p −
1) on PG(1 , p −
1) for Fermat prime p and a prime q dividing p − primitive graphs . We would like to remark that vertex-transitivegraphs of order pq , more precisely those with a primitive automorphism group and those with animprimitive automorphism group with blocks of size q , were also characterized by Praeger, Wangand Xu [60, 61, 67]. In fact, for the analysis of hamiltonian properties of the generalized orbitalgraphs arising from Row 5 of Table 3, a description from [61] will be used (see Section 6).Hamiltonicity of metacirculants and Fermat graphs of order pq has already been established,respectively, in [7, 46] and [50]. For the completion of the proof of Theorem 1.2 we need to provethe existence of Hamilton cycles for all the remaining graphs, that is, for primitive graphs oforder pq . Here we will use an approach which combines a variety of graph-theoretic and number-theoretic tools. For example, we will use the fact that the polycirculant conjecture has been settledfor certain vertex-transitive graphs. In particular, a vertex-transitive graph of order pq , p > q ,contains a fixed-point-free automorphism of order p . This allows an application of the lifting cycletechnique providing the quotient graph with respect to this automorphism of order p admits aHamilton cycle. When appropriate and when such automorphism exists, however, this techniquewill be applied to the quotient graph relative to a semiregular automorphism of order q . Theexistence of such a Hamilton cycle in the quotient is established using certain classical theoremson Hamilton cycles, such as the well-known Chv´atal’s theorem [15] and Jackson’s theorem [32],and also using some certain properties of finite fields. In particular, we obtain a novel result onpolynomials of degree 4 over finite fields of prime order with regards to a polynomial representationof quadratic residues at primitive roots, thus refining results from [43] (see Theorem 3.1). Thisresult will be used in Section 7 in all those cases for which Chv´atal’s theorem does not suffice toprove existence of a Hamilton cycle in the corresponding quotient.The proof of Theorem 1.2 is a lengthy analysis, covered in Sections 4, 5, 6 and 7, of hamiltonianproperties of all possible generalized orbital graphs arising from group actions in Table 3. In thesections preceding this analysis we fix the terminology and notation, gather same useful resultsand tools, and prove the above mentioned property of polynomials of degree 4 over finite fields.To be more concrete, the outline of this paper is given in the list of sections given below: over finite fields representing quadratic residues 94 Vertex-transitive graphs of order pq : explaining the strategy 165 Graphs arising from certain small rank/degree group actions 196 Actions of PSL(2 , q ) S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.2 Case S λ with λ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 PSL(2 , p ) S ξ with ξ = , S +0 ∪ S +1 and S − ∪ S − (Rows 3 and 4 of Table 4) . . . . . . . . . . . . . . . . . 377.3 Case S (Row 5 of Table 4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387.4 Cases S + and S − (Rows 6 and 7 of Table 4) . . . . . . . . . . . . . . . . . . . . . . . 397.5 Cases S + ξ and S − ξ with ξ = , Terminology, notation and some useful results
Throughout this paper graphs are finite, simple and undirected, and groups are finite, unlessspecified otherwise. Furthermore, a multigraph is a generalization of a graph in which we allowmultiedges and loops. Given a graph X we let V ( X ) and E ( X ) be the vertex set and the edge setof X , respectively. For adjacent vertices u, v ∈ V ( X ) we write u ∼ v and denote the correspondingedge by uv . The valency of a vertex u ∈ V ( X ) is denoted by val X ( v ) (or val ( v ) in short). If X is regular then its valency is denoted by val ( X ). Let U and W be disjoint subsets of V ( X ).The subgraph of X induced by U will be denoted by X h U i . Similarly, we let X [ U, W ] denote thebipartite subgraph of X induced by the edges having one endvertex in U and the other endvertexin W .Given a transitive group G acting on a set V , we say that a partition B of V is G - invariant if the elements of G permute the parts, the so-called blocks of B , setwise. If the trivial partitions { V } and {{ v } : v ∈ V } are the only G -invariant partitions of V , then G is primitive , and is imprimitive otherwise.A graph X is vertex-transitive if its automorphism group, denoted by Aut X , acts transitivelyon V ( X ). A vertex-transitive graph is said to be primitive if every transitive subgroup of itsautomorphism group is primitive, and is said to be imprimitive otherwise.A graph containing a Hamilton cycle will be sometimes referred as a hamiltonian graph. In this subsection we recall the orbital graph construction which is used throughout the rest ofthe paper. Orbital graphs can be constructed for any group action but in view of the fact that anytransitive action of a group G is isomorphic to the action of G on the coset space of a subgroupof G , we give this construction in the context of group actions on coset spaces. (In Section 6)we will, however, consider orbital graphs with respect to the action of PSL(2 , q ) ∼ = P Ω − (4 , q ) onparticular non-singular 1-dimensional vector spaces over GF( q ).)An action of a group G on the coset space G/H with respect to a subgroup H ≤ G givesrise to an action of G on G/H × G/H . Its orbits are called orbitals . Of course, the diagonal D = { ( x, x ) : x ∈ G/H } is always an orbital. Its complement, G/H × G/H − D is an orbital ifand only if G is doubly transitive. Unless specified otherwise, we only consider simply transitiveactions.An orbital is said to be self-paired if it simultaneously contains or does not contain orderedpairs ( x, y ) and ( y, x ), for x, y ∈ G/H . For an arbitrary union O of orbitals (not containing thediagonal D ), the generalized orbital (di)graph X ( G/H, O ) of the action of G on G/H with respectto O is a simple (di)graph with vertex set G/H and edge set O . (For simplicity reasons we willrefer to any such (di)graph as an orbital (di)graph of G .) It is an (undirected) graph if and onlyif O coincides with its symmetric closure, that is, O has the property that ( x, y ) ∈ O implies( y, x ) ∈ O . Further, the generalized orbital graph X ( G/H, O ) is said to be a basic orbital graph if O is a single orbital or a union of a single orbital and its symmetric closure.In terms of symmetry, the group G acts transitively on the vertices of X ( G/H, O ), and henceorbital (di)graphs are vertex-transitive. In fact, every vertex-transitive (di)graph can be con-structed in this way. 4he orbitals of the action of G on G/H are in 1-1 correspondence with the orbits of theaction of H on G/H , called suborbits of G . A suborbit corresponding to a self-paired orbital issaid to be self-paired . When presenting the (generalized) orbital (di)graph X ( G/H, O ) with thecorresponding (union) of suborbits S the (di)graph X ( G/H, O ) is denoted by X ( G, H, S ).In the example below the Petersen graph (the exceptional graph from Theorem 1.2) is describedas a basic orbital graph arising from the alternating group A . Example 2.1
Let G = A be the alternating group and let H = h (1 2) , (1 2 3) i be its subgroup.In the action of G on G/H × G/H there are two non-diagonal orbitals, both self-paired. Thecorresponding nontrivial suborbits of H are, respectively, of length and , giving the Petersengraph and its complement. Let m ≥ n ≥ ρ of a graph X is called ( m, n )- semiregular (in short, semiregular ) if as a permutation on V ( X ) it has a cycle decomposition consisting of m cycles of length n . If m = 1 then X is called a circulant ; it is in fact a Cayley graph of a cyclicgroup of order n . Let P be the set of orbits of ρ , that is, the orbits of the cyclic subgroup h ρ i generated by ρ . Let A, B ∈ P . By d ( A ) and d ( A, B ) we denote the valency of X h A i and X [ A, B ],respectively. (Note that the graph X [ A, B ] is regular.) We let the quotient graph correspondingto P be the graph X P whose vertex set equals P with A, B ∈ P adjacent if there exist vertices a ∈ A and b ∈ B , such that a ∼ b in X . We let the quotient multigraph corresponding to ρ be themultigraph X ρ whose vertex set is P and in which A, B ∈ P are joined by d ( A, B ) edges. Notethat the quotient graph X P is precisely the underlying graph of X ρ .The question whether all vertex-transitive graphs admit a semiregular automorphism is afamous open problem in algebraic graph theory (see, for example, [13, 19, 26, 27, 28, 45, 51, 66, 74]).A graph X admitting an ( m, n )-semiregular automorphism ρ can be represented, followingthe terminology established in [44], by an m × m array of subsets of H = h ρ i as well as withthe well-known Frucht’s notation [23]. For the sake of completeness we include both definitions.Let P = { S i : i ∈ Z m } be the set of m orbits of ρ , let u i ∈ S i and let S i,j be defined by S i,j = { t ∈ H : u i ∼ ρ t ( u j ) } . Then the m × m array ( S i,j ) is called the symbol of X relative to( ρ ; u , . . . , u m − ). To give a precise definition of Frucht’s notation let S i = { v ji | j ∈ Z n } where v i = u i and v ji = ρ j ( v i ). Then X may be represented in this notation by emphasizing the m orbitsof ρ in the following way. The m orbits of ρ are represented by m circles. The symbol n/R , where R ⊆ Z n \ { } , inside the circle corresponding to the orbit S i indicates that for each j ∈ Z n , thevertex v ji is adjacent to all the vertices v j + ri , where r ∈ R . When X h S i i is an independent set ofvertices we simply write n inside its circle. Finally, an arrow pointing from the circle representingthe orbit S i to the circle representing the orbit S k , k = i , labeled by the set T ⊆ Z n indicates thatfor each j ∈ Z n , the vertex v ji ∈ S i is adjacent to all the vertices v j + tk , where t ∈ T . An exampleillustrating this notation is given in Figure 1.We end this subsection with two important examples of graphs admitting semiregular auto-morphisms which arise in the classification of vertex-transitive graphs of order pq , see Section 4.The first class consists of the so-called metacirculant graphs, already mentioned in the introduc-tion as the class of vertex-transitive graphs of order pq admitting an imprimitive subgroup ofautomorphisms with blocks of size p , where p is the largest of the two primes p and q . A formaldefinition, first given in [6], goes as follows. An ( m, n )- metacirculant is a graph of order mn admitting an ( m, n )-semiregular automorphism ρ and an automorphism σ normalizing h ρ i which5yclically permutes the orbits of ρ . In particular, let the vertices of an ( m, n )-metacirculant X withrespect to an ( m, n )-semiregular automorphism ρ be denoted as in the previous paragraph whereFrucht’s notation is defined. Then X is uniquely determined by the array ( m, n, α, T , . . . , T µ )where α ∈ Z ∗ n , µ is the integer part of m/
2, the sets T i ⊂ Z n satisfy the following conditions0 / ∈ T = − T , α m T i = T i for 0 ≤ i ≤ µ , and if m is even, then α µ T µ = − T µ , the edge set of X is given by v ri ∼ v sj if and only if s − r ∈ α i T j − i , and the automorphism σ of X that normalizes h ρ i is defined by σ ( v ji ) = v αji +1 , where i ∈ Z m and j ∈ Z n . It can be easily seen that the semidirect product h ρ i ⋊ h σ i is a transitive subgroup of theautomorphism group of the metacirculant X . For example, a metacirculant given by the array(2 , , , {± } , { } ) is isomorphic to the Petersen graph.The second class consists of the so-called Fermat graphs [54], mentioned in the introductionas the class of vertex-transitive graphs of order pq admitting an imprimitive subgroup of auto-morphisms with blocks of size q and no imprimitive subgroup of automorphisms with blocks ofsize p , where p is the largest of the two primes p and q . In particular, let p = 2 s + 1 be aFermat prime and let q be a prime dividing p −
2. Let w be a fixed generator of the multiplicativegroup GF( p − ∗ = GF( p − \ { } of the Galois field GF( p − S of GF( q ) ∗ = GF( q ) \ { } and a non-empty proper subset T of GF( q ) ∗ we let the Fermat graph F ( p, q, S, T ) be the graph with vertex set PG(1 , p − × GF( q ) such that, for each point v of theprojective line PG(1 , p −
1) and each r ∈ GF( q ) ∗ , the neighbors of ( ∞ , r ) are all the vertices ofthe form ( ∞ , r + s ) ( s ∈ S ) and ( y, r + t ) ( y ∈ GF( p − , t ∈ T ) , and the neighbors of ( v, r ), v = ∞ , are all the vertices of the form( v, r + s ) ( s ∈ S ) and ( ∞ , r − t ) ( t ∈ T ) and ( v + w i , − r + t + 2 i ) ( i ∈ GF( q ) , t ∈ T ) . From this list of adjacencies one can easily see that F ( p, q, S, T ) admits a ( p, q )-semiregular auto-morphism, and that the quotient graph with respect to this automorphism is isomorphic to thecomplete graph K p . The smallest Fermat graph is the line graph of the Petersen graph. (Let usalso mention that the class of connected Fermat graphs is disjoint from the class of metacirculantsof order pq , see [54].) For group-theoretic terms not defined here we refer the reader to [68]. The following classicalgroups appear in the description of primitive group actions, of degree a product of two distinctprimes, which do not have imprimitive subgroups (see Table 3):(i) P Ω ǫ (2 d, d over a finitefield GF(2),(ii) PSL( n, q ): the projective special linear group on n -dimensional vector space over finite fieldof order q , 6iii) PGL( n, q ): the projective general linear group on n -dimensional vector space over finite fieldof order q ,(iv) M : the Mathieu group (a sporadic simple group of order 443520),(v) A n : the alternating group of degree n ,(vi) D n : the dihedral group of order 2 n . For a prime power r a finite field GF( r ) of order r will be denoted by F r , with the subscript r being omitted whenever the order of the field is clear from the context. The set of nonzeroquadratic residues modulo r , that is, elements of F ∗ = F \ { } that are congruent to a perfectsquare modulo r , will be denoted by S ∗ . The elements of S ∗ will be called squares , the elements of F ∗ not belonging to S ∗ will be called non-squares , and the set of all non-squares will be denotedby N ∗ , that is, N ∗ = F ∗ \ S ∗ .The following basic number-theoretic results will be needed throughout the paper. Proposition 2.2 [63, Theorem 21.2]
Let F be a finite field of odd prime order p . Then − ∈ S ∗ if p ≡ , and − ∈ N ∗ if p ≡ . Proposition 2.3 [63, Theorem 21.4]
Let F be a finite field of odd prime order p . Then ∈ S ∗ if p ≡ , , and ∈ N ∗ if p ≡ , . Proposition 2.4 [55, p. 167]
Let F be a finite field of odd prime order p . Then | S ∗ + 1 ∩ ( − S ∗ ) | = (cid:26) ( p − / p ≡ , ( p + 1) / p ≡ . In particular, if p ≡ then | S ∗ ∩ S ∗ + 1 | = ( p − / , | N ∗ ∩ N ∗ + 1 | = ( p − / , and | S ∗ ∩ N ∗ ± | = ( p − / . Using Proposition 2.4 the following result may be easily deduced.
Proposition 2.5
Let F be a finite field of odd prime order p . Then for any k ∈ F ∗ , the equation x + y = k has p − solutions if p ≡ , and p + 1 solutions if p ≡ . Proposition 2.6
Let F be a finite field of odd prime order p ≡ , and let A = S ∗ ∩ S ∗ +1 and B = S ∗ ∩ S ∗ − . Then | ( A \ B ) ∪ ( B \ A ) | ≥ , that is, | A ∪ B | ≥ | A | + 2 . Proof.
First observe that there must exist three consecutive elements s − , s, s + 1 of the field F such that s − s are squares but s + 1 is not. Therefore s ∈ S ∗ ∩ S ∗ + 1 but s / ∈ S ∗ ∩ S ∗ − s ∈ A \ B . But then, since − ∈ S ∗ , we have − s ∈ B \ A , and thus | ( A \ B ) ∪ ( B \ A ) | ≥ Proposition 2.7
Let F be a finite field of odd prime order p ≡ , and let A = S ∗ ∩ N ∗ +1 and B = S ∗ ∩ N ∗ − . Then | ( A \ B ) ∪ ( B \ A ) | ≥ , that is, | A ∪ B | ≥ | A | + 2 . Proof.
First observe that there must exist three consecutive elements s − , s, s + 1 of the field F such that s − s are squares but s + 1 is not. Therefore s ∈ S ∗ ∩ N ∗ − s / ∈ S ∗ ∩ N ∗ + 1,and so s ∈ B \ A . But then, since − ∈ S ∗ we have − s ∈ A \ B , and so | ( A \ B ) ∪ ( B \ A ) | ≥ roposition 2.8 Let F be a finite field of odd prime order p ≡ , let i = √− ∈ F ∗ ,and let T = { x ∈ F ∗ : 1 + x ∈ S ∗ } . Then for each x ∈ F ∗ at least one of the three elements x , ix and ix belongs to T . Proof.
If neither x nor ix are contained in T then both 1 + x and 1 + ( ix ) = 1 − x arenon-squares, and consequently 1 − x = (1 + x )(1 − x ) = 1 + ( ix ) is a square, and so ix belongs to T . In this subsection we list three results about existence of Hamilton cycles in particular graphsthat will prove useful in the subsequent sections. A recent result about the existence of Hamiltonpaths in those generalized Petersen graphs which do not have a Hamilton cycle is also given as itwill be needed in Subsection 7.5.
Proposition 2.9 [15] (Chv´atal’s Theorem)
Let X be a graph of order n . For a given positiveinteger i let S i = { x ∈ V ( X ) : deg( x ) ≤ i } . If for every i < n/ either | S i | ≤ i − or | S n − i − | ≤ n − i − then X contains a Hamilton cycle. Proposition 2.10 [32, Theorem 6] (Jackson’s Theorem)
Every -connected regular graph of order n and valency at least n/ contains a Hamilton cycle. The generalized Petersen graph GP ( n, k ) is defined to have the vertex set V ( GP ( n, k )) = { u i : i ∈ Z n } ∪ { v i : i ∈ Z n } , and edge set E ( GP ( n, k )) = { u i u i +1 : i ∈ Z n } ∪ { v i v i + k : i ∈ Z n } ∪ { u i v i : i ∈ Z n } . (1) Proposition 2.11 [2]
The generalized Petersen graph GP ( n, k ) , n ≥ and ≤ k ≤ n − containsa Hamilton cycle if and only if it is neither GP ( n, ∼ = GP ( n, n − ∼ = GP ( n, ( n − / ∼ = GP ( n, ( n + 1) / when n ≡ nor GP ( n, n/ when n ≡ and n ≥ . A graph is
Hamilton-connected if every pair of vertices is joined by a Hamilton path, and is
Hamilton-laceable if it is bipartite and every pair of vertices on opposite sides of the bipartitionis joined by a Hamilton path. The following result on existence of Hamilton paths in generalizedPetersen graphs without Hamilton cycles [62], will be needed later on.
Proposition 2.12 [62, Theorem 2.2]
Label the vertices of GP ( n, as in (1). Then the followinghold.(i) GP ( n, is Hamilton-connected if and only if n ≡ , .(ii) If n ≡ and x and y are distinct vertices of GP ( n, so that, for any i and t , { x, y } is neither of the pairs { u i , u i +2 } and { u i , u i +6 t } , then there is a Hamilton path joining x and y in GP ( n, .(iii) If n ≡ and { x, y } 6 = { v i , v i +4+6 t } , for any i and t , then there is a Hamilton pathjoining x and y in GP ( n, . iv) If n ≡ and { x, y } is none of the pairs { u i , u i +2 } , { u i , v i ± } , { u i , v i +2+6 t } , { v i , v i +4+6 t } ,for any i and t, then there is a Hamilton path joining x and y in GP ( n, .(v) If n ≡ , and x and y are not adjacent and { x, y } 6 = { v i , v i +3+6 t } for any i and t ,then there is a Hamilton path joining x and y in GP ( n, . over finite fields representing quadraticresidues In early eighties, motivated by a question posed by Alspach, Heinrich and Rosenfeld [5] in thecontext of decompositions of complete symmetric digraphs, Madden and V´elez [43] investigatedpolynomials that represent quadratic residues at primitive roots. They proved that, with finallymany exceptions, for any finite field F of odd characteristic, for every polynomial f ( x ) ∈ F [ x ] ofdegree r ≥ αg ( x ) or αxg ( x ) , there exists a primitive root β such that f ( β ) isa nonzero square in F . It is the purpose of this section to refine their result for polynomials ofdegree 4. This will then be used in Section 7 in the constructions of Hamilton cycles for some ofthe basic orbital graphs arising from the action of PSL(2 , p ) on cosets of D p − . This refinement,stated in the theorem below, will be proved following a series of lemmas. Theorem 3.1
Let F be a finite field of prime order p , where p is an odd prime not given inTables 1 and 2. Then for every polynomial f ( x ) ∈ F [ x ] of degree that has a nonzero constantterm and is not of the form αg ( x ) there exists a primitive root β ∈ F such that f ( β ) is a squarein F . The following result, proved in [43], is a basis of our argument and will be used throughoutthis section.
Proposition 3.2 [43, Corollary 1]
Let F be a finite field with p n elements. If s and t are integerssuch that(i) s and t are coprime,(ii) a prime q divides p n − if and only if q divides st , and(iii) φ ( t ) /t > rs − p n/ / ( p n −
1) + ( rs + 2) / ( p n − ,then, given any polynomial f ( x ) ∈ F [ x ] of degree r , square-free and with nonzero constant term,there exists a primitive root γ ∈ F such that f ( γ ) is a nonzero square in F . Throughout this section let p be an odd prime and let q = 2 , q , . . . , q m be the increasingsequence of prime divisors of p − q i q i · · · q i m m . As in [43] we define the following functionswith respect to this sequence: d ( n, m ) = 2(1 − q n )(1 − q n +1 ) · · · (1 − q m ) , (2) c r ( n, m ) = 2 r r ( q q · · · q n − q n q n +1 · · · q m ) , (3)and k ( m ) as the unique integer such that d ( k ( m ) − , m ) ≤ < d ( k ( m ) , m ). Hence k ( m ) ≥ d and c r can be defined for any positive integers r ≥ n < m andan arbitrary sequence { q , . . . , q m } of primes. The following lemma is a generalization of [43,Lemma 3]. 9 emma 3.3 Let { q , q , . . . , q m } be a finite sequence of primes satisfying m ≥ k ( m ) + 2 ,and let r = 4 . Then d ( k ( m ) + 1 , m ) − c r ( k ( m ) + 1 , m ) > . (4) Proof.
Since 2 ≤ k ( m ) ≤ m −
1, we have m ≥ . Let Ω be the increasing sequence of allprime numbers. For a given prime q , let I q = { w = 2 , w , w , . . . , w k ( m ) = q, w k ( m )+1 , . . . , w m } be a subsequence of Ω not missing any prime in Ω from the interval [ w , w m ]. Also, let d ( n, m ) ′ and c ( n, m ) ′ be the corresponding values for I q as defined by functions d and c r in (2) and (3).Further, let J q = { q = 2 , q , q , . . . , q k ( m ) = q, q k ( m )+1 , . . . , q m } denote the subsequence of Ω fromthe statement of Lemma 3.3. Then one can easily see that d ( k ( m ) + 1 , m ) ′ ≥ d ( k ( m ) + 1 , m ) andthat c ( k ( m ) + 1 , m ) ′ ≤ c ( k ( m ) + 1 , m ), and so (4) holds for J q if it holds for I q . This showsthat in what follows we can assume that J q = I q .Since d ( k ( m ) + 1 , m ) = (1 + 1 w k ( m ) − d ( k ( m ) , m ) > w k ( m ) − , (4) holds if 1 + 1 w k ( m ) − − r ( w w · · · w k ( m ) w k ( m )+1 w k ( m )+2 · · · w m ) > , which may be rewritten in the following form w w · · · w k ( m ) ( w k ( m ) − < w k ( m )+1 · · · w m − w m , (5)in view of the fact that r = 4 and w = 2.We divide the proof into two cases, depending on whether m ≥ m = 6. Case 1. m ≥ w w · · · w l ( w l − < w l +1 · · · w m − w m , (6)where m ≥ l ≤ m − w m ≥ m − l ) − ( l − m − l − ≥ , (7)then (6) holds provided w m − w m >
128 holds. Note that this is true if w m ≥
13, which is the casesince m ≥
7. Next, note that for either m being even and l < m − m being odd, (7) holds.So we may assume that m is even and that l = m/ − ≥ . Now we prove that (6) holds under this assumption for any even integer m ≥ m = 8. Then l = 3 and (6) rewrites as w w ( w − < w w w w w . (8)A computer search shows that (8) holds for all primes w ≤ m ≥
8. Then we have w w w · · · w l w l +1 ( w l +1 − = w ( w · · · w l w l +1 ( w l +1 − ) < w ( w l +2 w l +3 · · · w m w m +1 ) < ( w l +2 w l +3 · · · w m w m +1 ) w m +2 . m ≥ m ≥ . Hence (5) holds,and so does (4).
Case 2. m = 6.Then k ( m ) = 2, and so (6) becomes w ( w − < w w w w . (9)A computer search shows that (9) does not hold only for w k ( m ) = w ∈ { , , , , , , , , , , , , , , } . An additional computer search shows that for w = 2 (4) holds in each of these exceptional cases.This completes the proof of Lemma 3.3.The following result proved in [43] will be needed in the next lemma. Proposition 3.4 [43, Lemma 5]
Let { q , q , . . . , q m } be a finite sequence of primes satisfying m ≤ k ( m ) + 1 . Then m ≤ and q k ( m ) − ≤ . In fact the sequence must satisfy one of thefollowing:(i) k ( m ) = 4 , q k ( m ) − = 5 and m = 9 ,(ii) k ( m ) = 3 , q k ( m ) − = 5 and m ≤ ,(iii) k ( m ) = 3 , q k ( m ) − = 3 and m ≤ , or(iv) k ( m ) = 2 , q k ( m ) − = 2 and m ≤ . Lemma 3.5
Let { q , q , . . . , q m } be a finite sequence of primes satisfying m ≤ k ( m ) + 1 ,and let p − q i q i · · · q i m m with q m ≥ . Then there exist s and t such that(i) s and t are coprime,(ii) a prime q divides p − if and only if q divides st , and(iii) φ ( t ) /t > s − √ p/ ( p −
1) + (4 s + 2) / ( p − . Proof.
Since m ≤ k ( m ) + 1 the four cases (i) - (iv) of Proposition 3.4 need to be considered. Ineach case, as in [43, Lemma 7], we will prescribe a choice for s (which then determines t uniquely)and use the conditions in each of these four cases to find the lower bound α for the expression(2 φ ( t ) t − − φ ( t ) t − − ≥ α . We will then be able to use the assumption q m ≥ α > (4 s − √ p + 4 s + 2 p − . (10)Suppose first that Proposition 3.4(i) holds, that is, k ( m ) = 4, q k ( m ) − = 5 and m = 9. Then q ≥ q = 2, q = 3and q = 5. Let s = 2 · · t = q q · · · q . Then2 φ ( t ) t − ≥ −
17 )(1 −
111 )(1 −
113 )(1 −
117 )(1 −
119 )(1 − − ≥ . . p satisfies (10) with α = 0 . s = 30 if and only if p > p ≤ · · · q divides p − q ≥ q q q q q < / (2 · · · ≤ q q q q q ≥ · · · ·
19 = 323323.We now consider the other three cases of Proposition 3.4, that is, suppose that Proposi-tion 3.4(ii), (iii) or (iv) holds. In all three cases k ( m ) ≤
3. Since p is an odd prime we know q = 2, and we now consider the various possibilities for q . First, assume that q = 3 (note thatthis is possible in the last two cases) and therefore m ≤
7. We set s = 2 · t = q q q q q .Thus 2 φ ( t ) t − ≥ −
15 )(1 −
17 )(1 −
111 )(1 −
113 )(1 − − ≥ . . Now p satisfies (10) with α = 0 . s = 6 if and only if p ≥ p < q q · · · q m − < / (2 · · <
31. Since q i ≥ i ∈ { , , . . . , m − } one can seethat either m = 3 or m = 4. In other words, either t = q or t = q q , and thus we can improvethe value for α with 2 φ ( t ) t − ≥ −
15 )(1 − − ≥ . . In this case p satisfies (10) with α = 0 . p > p ≤ q m divides p − q m ≥
131 implies that q <
2, a contradiction.We now use the same approach for the case q = 5. We choose s = 2 · t = q q · · · q m .Here we have2 φ ( t ) t − ≥ −
17 )(1 −
111 )(1 −
113 )(1 −
117 )(1 − − ≥ . . Hence p satisfies (10) with α = 0 . p > p ≤ q m divides p − q <
10, and so either m = 4 and q = 7 or m = 3. In both caseswe can improve the value for α since t = q or t = q q . In particular,2 φ ( t ) t − ≥ −
17 )(1 − − ≥ . . In this case p satisfies (10) with α = 0 . p > p ≤ q m divides p − q m ≥
131 implies that q <
2, a contradiction.Finally we consider the case q ≥
7. Then, by Proposition 3.4, we have k ( m ) = 2 and m ≤ s = 2 and use the same technique as above to complete the proof. In particular,we have 2 φ ( t ) t − ≥ −
17 )(1 −
111 )(1 −
113 )(1 − − ≥ . . In this case p satisfies (10) with α = 0 . p > p ≤
243 observe that theassumption that 2 q m divides p − q m ≥
131 implies that q <
2, a contradiction.In summary we have seen that given any finite sequence of primes with q m ≥
131 we canchoose n in such a way that when s = q q · · · q n and t = q n +1 q n +2 · · · q m we have2 φ ( t ) t > s − √ st + 1 st + 4 s + 2 st , (11)completing the proof of Lemma 3.5. 12n order to proceed with the proof of Theorem 3.1 we now need to identify all those se-quences { q , q , . . . , q m } with q m <
131 for which one cannot choose s = q q · · · q n and t = q n +1 q n +2 · · · q m so as to satisfy (11). Since Lemma 3.3 holds for each q m we can assume thatfor each of these sequences Proposition 3.4 applies. A computer search of these finitely manysequences yields the exceptional sequences which are listed in Tables 1 and 2. For each of theseexceptional sequences we fix s = q q · · · q n and t = q n +1 q n +2 · · · q m , and we then search for aconstant k such that x > k implies the inequality2 φ ( t ) t > s − √ xx − s + 2 x − . (12)For each of these sequences Tables 1 and 2 give the smallest bound k obtained in this way. Thethird column of these tables indicates for which choice of t the given bound k is obtained: Type1 means that the bound k was obtained with t = q m − q m , Type 2 means that the bound wasobtained with t = q m , and Type 3 means that the bound was obtained with t = 1. A computersearch then identifies those primes that are smaller than or equal to the bound k , as summarizedin the proposition below. Proposition 3.6
Let { q , q , . . . , q m } be a finite sequence of primes satisfying m ≤ k ( m )+1 ,and let p − q i q i · · · q i m m with q m < . If p is not listed in Tables 1 and 2 then there exist s and t such that(i) s and t are coprime,(ii) a prime q divides p − if and only if q divides st , and(iii) φ ( t ) /t > s − √ p/ ( p −
1) + (4 s + 2) / ( p − . We are now ready to prove Theorem 3.1.
Proof of Theorem 3.1.
It follows by Proposition 3.2 that a polynomial f ( x ) represents anonzero square at some primitive root in F if there exist s and t satisfying the following threeconditions:(i) s and t are coprime,(ii) a prime q divides p − q divides st , and(iii) 2 φ ( t ) /t > s − √ p/ ( p −
1) + (4 s + 2) / ( p − s and t exist for all odd primes p that are not listed inTables 1 and 2.Let { q = 2 , q , . . . , q m } be an increasing sequence of prime divisors of p −
1. If m ≤ k ( m ) + 1then Lemma 3.5 applies for q m ≥ q m < m ≥ k ( m ) + 2. Then, by Lemma 3.3, we have d ( k ( m ) + 1 , m ) > c ( k ( m ) + 1 , m ) . If we let s = q q · · · q k ( m ) and t = q k ( m )+1 · · · q m we have 2 φ ( t ) /t = d ( k ( m ) + 1 , m ), and c ( k ( m ) + 1 , m ) = 8 · s q q · · · q k ( m ) q k ( m )+1 q k ( m )+2 · · · q m = 8 s √ q q · · · q m ≥ s √ p − T k Type p ≤ k p ≡ ≤ k with T with T , ( p + 1) / , ,
17 52 , , ,
11 2458 1 331 , , , , , , ,
43 1622 1 1291 no , , ,
17 1372 1 no no , , , ,
13 7040 t = 455 2731 no , ,
43 460 1 no no , ,
31 496 1 373 no , ,
61 435 1 367 no , , , ,
23 5145 t = 805 4831 no , ,
23 547 1 139 ,
277 2772 , ,
67 430 1 no no , , ,
13 1517 1 547 , , ,
17 632 1 103 , , ,
613 6132 , , ,
13 2238 1 1171 , no , ,
11 788 2 67 , , ,
727 3972 , no , ,
13 739 2 79 , ,
313 157 , , , , , , , , , , , ,
23 65 2 47 no , , ,
37 1656 1 no no , , , no , , ,
41 1632 1 1231 no , ,
59 437 1 no no , ,
53 444 1 no no , , ,
19 1327 1 no no , , ,
29 1727 1 no no ,
17 69 2 no no ,
11 78 2 23 no , , ,
19 1921 1 571 no , ,
41 464 1 no no
Table 1: The list of sequences not satisfying (11) I.Since s is even and 4( p − ≥ s ≥ s − √ pp − ≤ s √ p − . It follows that 2 φ ( t ) t = d ( k ( m ) + 1 , m ) ≥ c ( k ( m ) + 1 , m ) ≥ s √ p − ≥ s − √ pp − s + 2 p − . T k Type p ≤ k p ≡ ≤ k with T with T , ( p + 1) / , , , ,
11 8160 t = 385 2311 , , , , , , , , , , , , , , , , , ,
47 1604 1 no no , , ,
31 1705 1 no no , , ,
23 1265 1 967 no , ,
17 180 1 no no , , ,
13 1130 1 859 no ,
13 74 2 53 no , ,
11 218 1 no no , ,
13 200 1 131 no , ,
37 475 1 223 no , , , , , , , , , , , , ,
19 36145 1 11971 , no , , , , , , , , ,
193 13 , , , , , , no , , ,
23 1819 1 691 , , ,
47 453 1 283 no , , , ,
17 37400 1 3571 , , , no , ,
29 506 1 349 no , , ,
11 1646 1 463 no , , ,
17 1995 1 1021 , no ,
29 63 2 59 no , ,
19 596 1 229 ,
457 4572 ,
19 68 2 no no
Table 2: The list of sequences not satisfying (11) II.(Note that the last inequality hold since p ≥ , p ) on the cosets of D p − the following result about particular polynomials over finite fields of prime order p , where p is one of the primes listed in the last column of Tables 1 and 2, obtained with a computer search,will be needed. Proposition 3.7
Let F be a finite field of odd prime order p , and let k ∈ F . If p ∈ { , , , , , , , , , , , , , , , , , , , , , } then there exists a primitive root β of F such that f ( β ) = β + kβ + 1 is a square in F except hen ( p, k ) ∈ { (5 , , (13 , , (13 , , (13 , , (13 , , (13 , , (13 , , (37 , , (37 , , (37 , , (61 , , (61 , , (61 , } . Amongst these exceptions only for ( p, k ) ∈ { (13 , , (37 , , (61 , } there exists ξ ∈ S ∗ ∩ S ∗ +1 such that k = 2(1 − ξ ) . In particular, ξ = 10 for ( p, k ) = (13 , , ξ = 12 for ( p, k ) =(37 , , and ξ = 57 for ( p, k ) = (61 , . Moreover, amongst these exceptions only for ( p, k ) ∈{ (13 , , (37 , , (61 , } there exists ¯ ξ ∈ S ∗ ∩ S ∗ + 1 such that k = − − ξ ) . In particular, ¯ ξ = 4 for ( p, k ) = (13 , , ¯ ξ = 26 for ( p, k ) = (37 , , and ¯ ξ = 5 for ( p, k ) = (61 , . pq : explaining the strategy The goal of this paper is to prove that the Petersen graph is the only connected vertex-transitivegraph of order a product of two primes without a Hamilton cycle. Recall that vertex-transitivegraphs of prime-squared order are necessarily Cayley graphs of abelian groups. The existence ofHamilton cycles in such graphs was proved by the third author in 1983 [47]. If one of the twoprimes is equal to 2 then the graphs are of order twice a prime, and the existence of Hamiltoncycles in such graphs (with the exception of the Petersen graph) was proved by Alspach back in1979 [1].
Proposition 4.1 [1, 47]
Let p be a prime. With the exception of the Petersen graph, everyconnected vertex-transitive graph of order qp , where q ∈ { , p } , contains a Hamilton cycle. In pursuing our goal we therefore only need to consider vertex-transitive graphs whose orderis a product of two different odd primes p and q ( p > q ). As mentioned in the introduction thereare three disjoint classes of such graphs. Recall that the first class consists of graphs admittingan imprimitive subgroup of automorphisms with blocks of size p - it coincides with the class of( q, p )-metacirculants defined in Subsection 2.3. The second class consists of graphs admittingan imprimitive subgroup of automorphisms with blocks of size q and no imprimitive subgroupof automorphisms with blocks of size p - it coincides with the class of Fermat graphs defined inSubsection 2.3. Finally, the third class consists of vertex-transitive graphs with no imprimitivesubgroup of automorphisms. Following [58, Theorem 2.1] the theorem below gives a completeclassification of connected vertex-transitive graphs of order pq . We would like to remark, however,that there is an additional family of primitive graphs of order 91 = 7 ·
13 that was not coveredneither in [58] nor in [61]. This is due to a missing case in Liebeck - Saxl’s table [41] of primitivegroup actions of degree mp , m < p . This missing case consists of primitive groups of degree91 = 7 ·
13 with socle PSL(2 ,
13) acting on cosets of A . In the classification theorem below thismissing case is included in Row 7 of Table 3. Theorem 4.2 [58, Theorem 2.1]
A connected vertex-transitive graph of order pq , where p and q are odd primes and p > q , must be one of the following:(i) a metacirculant,(ii) a Fermat graph,(iii) a generalized orbital graph associated with one of the groups in Table 3. soc G ( p, q ) action comment P Ω ǫ (2 d,
2) (2 d − ǫ, d − + ǫ ) singular ǫ = +1 : d Fermat prime1-spaces ǫ = − d − M (11 ,
7) see Atlas3 A (7 ,
5) triples4 PSL(2 ,
61) (61 ,
31) cosets of A , q ) ( q +12 , q ) cosets of q ≥ , q )6 PSL(2 , p ) ( p, p +12 ) cosets of p ≡ mod D p − p ≥
137 PSL(2 ,
13) (13 ,
7) cosets of missing in [41] A Table 3:
Primitive groups of degree pq without imprimitive subgroups and with non-isomorphic generalizedorbital graphs. The existence of Hamilton cycles in graphs given in Theorem 4.2(i) and (ii) was proved,respectively, in [7] and [50]. For the sake of completeness, let us briefly explain the correspondingconstruction methods.In [7] the existence of Hamilton cycles was proved for all ( m, n )-metacirculants with m odd,and not only for metacirculants of order a product of two odd primes. Let, however, X be ametacirculant defined by the array ( q, p, α, T , . . . , T µ ) as in Subsection 2.3. If gcd( c, q ) = 1, where c = a/ gcd( a, q ) and a is the order of α ∈ Z ∗ p , then it follows by [6] that X is a Cayley graph ofthe group h ρ, σ c i = h ρ i ⋊ h σ c i . Thus in this case the result about existence of Hamilton cycles inCayley graphs of semidirect products of a prime order cyclic group with an abelian group, provedin [20, 47], can be applied. When gcd( c, q ) = 1 one can use the fact that the quotient X P withrespect to the set of orbits P of ρ , is a circulant of order q with symbol {± i : 1 ≤ µ, and T i = ∅} .A Hamilton cycle in X is then constructed as a lift of a particular Hamilton cycle in X P (see [7]for details).Hamilton cycles in Fermat graphs were constructed in [50] in the following way. Let X = F ( p, q, S, T ) be a Fermat graph with p = 2 s + 1 being a Fermat prime and q a prime dividing p − p − s − p, q ) = (5 , X is isomorphic to the line graph of thePetersen graph which clearly admits a Hamilton cycle. It can therefore be assumed that q < p − X Q with respect to the set of orbits Q of a ( p, q )-semiregular automorphism in X is isomorphic to the complete graph K p . Then in X Q a particular ( p − { ( ∞ , r ) : r ∈ F q } is constructed in such a way that it lifts to a cycle C in the originalgraph X leaving out only vertices of the form ( ∞ , r ), r ∈ F q . One can then show that each ofthese missing vertices is adjacent to two neighboring vertices on C . The cycle C can therefore beextended to a Hamilton cycle in X (see [50] for details).Combining the above results from [7, 50] with Proposition 4.1 we have a complete solutionof the hamiltonicity problem for vertex-transitive graphs of order a product of two primes in the17mprimitive case. Proposition 4.3
With the exception of the Petersen graph, every connected vertex-transitivegraph of order qp , where p and q are primes, with an imprimitive subgroup of automorphismscontains a Hamilton cycle. In view of Proposition 4.3 it follows that Theorem 1.2 will be proved and our goal will beachieved if we manage to show that every primitive graph of order pq contains a Hamilton cycle.More precisely, we need to show that graphs arising from primitive group actions given in Table 3have a Hamilton cycle. The existence of Hamilton cycles needs to be proved for all connectedgeneralized orbital graphs arising from these actions (see Subsection 2.2). Recall that a generalizedorbital graph is a union of basic orbital graphs. Since the above actions are primitive and hencethe corresponding basic orbital graphs are connected, it suffices to prove the existence of Hamiltoncycles solely in basic orbital graphs of these actions. This is done in the next three sections.Graphs arising from the actions in the first four rows and the last row of Table 3 are consideredin Section 5. Existence of Hamilton cycles in these graphs is proved using Jackson’s theorem (seeProposition 2.10) and in some cases combined also with an ad hoc computer based search.Graphs arising from the actions in Rows 5 and 6 are considered, respectively, in Sections 6and 7. The method used in the proof of existence of Hamilton cycles in these graphs is for themost part based on the so-called lifting cycle technique [3, 37, 47]. Lifts of Hamilton cycles fromquotient graphs which themselves have a Hamilton cycle are always possible, for example, whenthe quotienting is done relative to a semiregular automorphism of prime order and when thecorresponding quotient multigraph has two adjacent orbits joined by a double edge contained ina Hamilton cycle. This double edge gives us the possibility to conveniently “change direction”so as to get a walk in the quotient that lifts to a full cycle above (see Example 4.4). We remarkthat by [45] a vertex-transitive graph of order pq , q < p primes, contains a ( q, p )-semiregularautomorphism. Consequently the lifting cycle technique can be applied to graphs arising fromRows 5 and 6 of Table 3 provided appropriate Hamilton cycles can be found in the correspondingquotients. Note, however, that in graphs arising from Row 5 of Table 3 the quotienting is done withrespect to a ( p, q )-semiregular automorphism (see Section 6). As one would expect it is preciselythe existence of such Hamilton cycles in the quotients that represent the hardest obstacle oneneeds to overcome in order to assure the existence of Hamilton cycles in the graphs in question.In this respect many specific tools will be applied, most notably the classical Chv´atal’s theorem[15], and results on polynomials from Section 3.In the example below we illustrate this method on one of the basic orbital graphs of valency6 arising from Row 6 of Table 3. Example 4.4 [37, Example 3.2] The orbital graph X arising from the action of PSL(2 ,
13) oncosets of D with respect to a union ( S +7 ∪ S − from Example 7.3) of a suborbit of size 3 and itspaired suborbit contains a (7 , ρ , and it can be nicely representedin Frucht’s notation as shown in Figure 1. Since the quotient graph X ρ has a Hamilton cyclecontaining a double edge and since 13 is a prime, this cycle lifts to a Hamilton cycle in the originalgraph X (see Figure 1). Example 4.5
The so-called odd graph O is a basic orbital graph of valency 4 arising fromthe action of the alternating group A acting on 3-subsets of { , . . . , } where two 3-subsets areadjacent if they are disjoint. The odd graph O has a (5 , A vertex-transitive graph arising from the action of PSL(2 ,
13) on cosets of D given in Frucht’s notationwith respect to the (7 , ρ where undirected lines carry label 0. Edges in bold show aHamilton cycle. (7 , O is hamiltonian. In the below matrix we also give the symbol of O with respect to the orbits S i = { v ji : j ∈ Z } , i ∈ Z , of a (7 , ∅ { } ∅ ∅ { } ∅ { , }{ } ∅ { , } ∅ ∅ ∅ { }∅ { , } ∅ { , } ∅ ∅ ∅∅ ∅ { , } ∅ { } { } ∅{ } ∅ ∅ { } ∅ { , } ∅∅ ∅ ∅ { } { , } ∅ { }{ , } { } ∅ ∅ ∅ { } ∅ . It will be useful to introduce the following terminology. Let X be a graph that admits an( m, n )-semiregular automorphism ρ . Let P = { S , S , . . . , S m } be the set of orbits of ρ , and let π : X → X P be the corresponding projection of X to its quotient X P . For a (possibly closed)path W = S i S i . . . S i k in X P we let the lift of W be the set of all paths in X that project to W .The proof of following lemma is straightforward and is just a reformulation of [52, Lemma 5]. Lemma 4.6
Let X be a graph admitting an ( m, p ) -semiregular automorphism ρ , where p is aprime. Let C be a cycle of length k in the quotient graph X P , where P is the set of orbits of ρ .Then, the lift of C either contains a cycle of length kp or it consists of p disjoint k -cycles. In thelatter case we have d ( S, S ′ ) = 1 for every edge SS ′ of C . We deal here with the first four rows and the last row of Table 3 of which the first three rows19
555 5 5 5
Figure 2:
The odd graph O given in Frucht’s notation with respect to a (7 , ρ whereundirected lines carry label 0. correspond to groups of rank 3 and 4.In the first proposition we show the existence of Hamilton cycles in the graphs arising fromthe first three rows of Table 3. With the exception of one of the graphs arising from the action of M (Row 2 of Table 3) and of the odd graph O (Row 3 of Table 3, see Example 4.5) for whichthe Hamilton cycle is constructed via the lifting Hamilton cycle technique, in all other cases thehamiltonicity is proved using Proposition 2.10. Proposition 5.1
Vertex-transitive graphs arising from primitive groups in Rows 1-3 of Table 3are hamiltonian.
Proof.
Consider first the rank 3 action of the orthogonal group P Ω ǫ (2 d,
2) on singular 1-spaces,for ǫ ∈ { +1 , − } , in Row 1 of Table 3. The primes p and q are as follows: p = 2 d − q = 2 d − + 1 for P Ω + (2 d,
2) and p = 2 d + 1 and q = 2 d − − P Ω − (2 d, d − − d − + 2) = q − q − d − = q − q + 1.Similarly, in the second case the valencies of the two graphs are (2 d − + 1)(2 d − −
2) = q + q − d − = q + 2 q + 1. It is straightforward to see that for each of these graphs the valencyexceeds one third of the corresponding order, and so, by Proposition 2.10, the result follows.Consider now the action of M of rank 3 and degree 77 given in Row 2 of Table 3. Thecorresponding nontrivial suborbits are of cardinalities 16 and 60. Again Proposition 2.10 appliesto the graph of valency 60. As for the graph of valency 16 we observe that the quotieningrelative to a (7 , K and contains multiple edges. An appropriate Hamilton cycle is then chosenand Lemma 4.6 applied to obtain a Hamilton cycle in the original graph.Finally, consider A acting on triples in { , , . . . , } from Row 3 of Table 3. This action givesrise to six different graphs associated with three nontrivial suborbits of cardinalities 4, 12 and 18.Clearly, in view of Proposition 2.10, there is a Hamilton cycle in each of these graphs with theexception of the odd graph O of valency 4. As for the Hamilton cycle in O it is constructed in20xample 4.5. We would like to remark, however, that the hamiltonicity of O was first proved byBalaban in [9].The generalized orbital graphs arising from the action of PSL(2 ,
61) on the cosets of its maximalsubgroup isomorphic to A (Row 4 of Table 3) are of order 1891 = 31 ·
61. This action has 40nontrivial suborbits, 32 of which are self-paired and 8 are not self-paired. More precisely, onesuborbit is of length 6, one of length 10, two of length 12, four of length 20, five of length 30 and27 of length 60. Of the latter 8 are non-self-paired. In Figure 3 the graph of valency 6 is showntogether with a Hamilton cycle in the quotient graph with respect to a semiregular automorphismof order 61 that, by Lemma 4.6, lifts to a full Hamilton cycle in the graph itself. In a similarmanner Hamilton cycles are constructed in the remaining orbital graphs. Also, hamiltonicity ofall of these 36 basic orbital graphs has been checked using Magma [11]. Hence the followingproposition holds.
Proposition 5.2
Vertex-transitive graphs of order · arising from the action of PSL(2 , on A given in Row 4 of Table 3 are hamiltonian. We end this section with generalized orbital graphs of order 91 = 7 ·
13 arising from Row 7in Table 3. It was proved in [59] that the subdegrees of this action are 1, 4, 4, 4, 6, 12, 12, 12,12, 12, 12. With the exception of two suborbits of length 12 all other suborbits are self-paired.The corresponding basic orbital graphs are therefore of the following valencies: three of valency4, one of valency 6, four of valency 12 and one of valency 24. Using Magma [11] it can be checkedthat up to isomorphism there are in fact only two graphs of valency 4 and only three graphs ofvalency 12. For each of these seven graphs we find a semiregular automorphism whose quotientcontains a Hamilton cycle that lifts to a Hamilton cycle in the original graph. In all cases, withthe exception of one of the graphs of valency 4 where this automorphism is (13 , Proposition 5.3
Vertex-transitive graphs of order
91 = 7 · arising from the action of PSL(2 , on A given in Row 7 of Table 3 are hamiltonian. PSL(2 , q ) In this section the existence of Hamilton cycles in basic orbital graphs arising from the groupaction PSL(2 , q ) on the cosets of PGL(2 , q ) given in Row 5 of Table 3 is considered. The followinggroup-theoretic result due to Manning will be needed in this respect. Proposition 6.1 [68, Theorem 3.6’]
Let G be a transitive group on Ω and let H = G α for some α ∈ Ω . Suppose that K ≤ G and at least one G -conjugate of K is contained in H . Suppose furtherthat the set of G -conjugates of K which are contained in H form t conjugacy classes of H withrepresentatives K , K , · · · , K t . Then K fixes P ti =1 | N G ( K i ) : N H ( K i ) | points of Ω . Let G = PSL(2 , q ), where q ≥ G has two conjugacy classes ofsubgroups isomorphic to PGL(2 , q ), with the corresponding representatives H and H ′ . Since each21 v v v v v v v v v v v v v v v v v v v u u u u u u u u u u u u u u u
38 8 47 56 24350 26 21 21 345 37 33 28 132431334245 42 2 11 13 7 415135353621393715 22 30 30 402 40532940 13 40 23 38 4 56414 1616 2225 9 17 5 55 3251650 15 25524 12 2215523928 40620
29 10263017 28 242311 8 6 21182 7
Figure 3:
The basic orbital graph X arising from the action of PSL(2 ,
61) on the cosets of A of valency 6 givenin Frucht’s notation where labels inside circle are omitted if there are no edges inside orbits and where we only givethe label a inside the circle if the corresponding orbit induces a cycle with jumps a (circles in bold). A Hamiltoncycle in X ρ that lifts to a Hamilton cycle in X is presented with bold grey lines. element in PGL(2 , q ) interchanges these two classes, it suffices to consider the action of G on theset H of right cosets of H in G . The degree of this action is pq , where p = ( q + 1) /
2. Further, let P be a subgroup of H ′ of order q , that is, a subgroup of G of order q which has trivial intersectionwith H . We have the following result. Lemma 6.2
The action of P on H is semiregular. Furthermore, the action of its normalizer N G ( P ) on H has q +12 orbits of length q and one orbit of length q ( q − . Proof.
We first prove that the action of P on H is semiregular. Suppose on the contrary thatthere exists g ∈ G such that HgP = Hg . Then HgP g − = H , and so gP g − ≤ H . But thiscontradicts the choice of P . Hence P is semiregular on H .22 /2 7 7 77777777 7 7
13 13 13131313 13
Figure 4:
The two vertex-transitive graphs arising from the action of PSL(2 ,
13) on cosets of A of valency 4 givenin Frucht’s notation with respect to, respectively, the (13 , , We now compute the orbits of the normalizer N = N G ( P ) ∼ = Z q ⋊ Z q − of P in G in its actionon H , by analyzing subgroups of H conjugate in G to subgroups of N . (Note that there is onlyone conjugacy class of subgroups in G isomorphic to N .) A subgroup of N is isomorphic to oneof the following groups Z q ⋊ Z q − , Z q , Z q ⋊ Z q − , Z q ⋊ Z l , where 2 ≤ l < q −
1, and Z l , where l divides q − N cannot fix a coset in H for otherwise there would exists g ∈ G such that HgN = Hg , and so gN g − ≤ H which is impossible since | N | = q ( q −
1) and | H | = ( q − q ( q + 1). Also,as P is semiregular on H no subgroup of N isomorphic to Z q fixes a coset in H .The group N contains q + 1 maximal subgroups isomorphic to Z q ⋊ Z q − , which form q + 1different conjugacy classes in N , but are divided into two equal size classes in G , one containingsubgroups of H and the other containing subgroups of H ′ . Each of these two conjugacy classescontains q +12 subgroups of N . Let K be such a subgroup of H isomorphic to Z q ⋊ Z q − . Since N G ( K ) = N H ( K ) = K , it follows, by Proposition 6.1, that K fixes only the coset H . Hence anysubgroup of N conjugate to K in G fixes one coset of H , and the corresponding orbit of N on H is of length | N | / | K | = q . Since N admits q +12 subgroups conjugate to K in G , which form q +12 different conjugacy classes inside N , we can conclude that N has q +12 orbits of length q .A subgroup K ≤ K of H isomorphic to Z q ⋊ Z l , where 2 ≤ l < q −
1, has the same fixedcosets as K (and so it is a subgroup of a coset stabilizer). Consequently N does not have orbits oflength q · q − l for 2 ≤ l < q −
1. Further, for any subgroup K ≤ K of H isomorphic to Z l , where l divides q − l = 2, the fact that | N G ( K ) : N H ( K ) | = | D q − : D q − | = q +12 , implies that K fixes q +12 cosets. These cosets are clearly contained in the above q +12 orbits of N of length q ,and consequently N does not have orbits of length q − l . We have therefore show that the only other possible stabilizers are Z and Z . Since |H| = q ( q + 1) / N on H with coset stabilizer isomorphic to Z or23o Z equals, respectively, q ( q − and q ( q − q ( q + 1)2 = q q + 12 + a q ( q − bq ( q − , (13)where a is the number of orbits of N on H with coset stabilizer isomorphic to Z and b is thenumber of orbits of N on H on which N acts regularly. The equation (13) simplifies to q = q + aq ( q −
1) + 2 bq ( q − , which clearly has a = 1 and b = 0 as the only possible solution. This completes the proof ofLemma 6.2.Lemma 6.2 will play an essential part in our construction of Hamilton cycles in basic orbitalgraphs arising from the action of PSL(2 , q ) on cosets of PGL(2 , q ) given in Row 5 of Table 3.The strategy goes as follows. Let X be such an orbital graph. By Lemma 6.2, the action of thenormalizer N = N G ( P ) on the quotient graph X P with respect to the orbits P of a semiregularsubgroup P consists of one large orbit of length q ( q − / q + 1) / X by first showing that the subgraph of X P inducedon the large orbit has at most two connected components and that each component contains aHamilton cycle with double edges in the corresponding quotient multigraph. If the component isjust one then its Hamilton cycle is modified to a Hamilton cycle in X P by choosing in an arbitrarymanner ( q + 1) / q + 1) / N in X P . By Lemma 4.6, this cycle lifts to a Hamilton cycle in X . Such2-paths indeed exist because every isolated vertex has to be adjacent to every vertex in the largeorbit (see Lemma 6.5). If the subgraph of X P induced on the large orbit has two componentswith corresponding Hamilton cycles C and C , then a Hamilton cycle in X is constructed by firstconstructing a Hamilton cycle in X P in the following way. We use two isolated vertices to modifythese two cycles C and C into a cycle of length q ( q − / C andan edge in C by two 2-paths each having one endvertex in C and the other in C , whereas thecentral vertices are the above two isolated vertices. In order to produce the desired Hamilton cyclein X P the remaining isolated vertices are attached to this cycle in the same manner as in the caseof one component. By Lemma 4.6, this cycle lifts to a Hamilton cycle in X . Formal proofs aregiven in Propositions 6.7 and 6.8.It follows from the previous paragraph that we only need to prove that the subgraph of X P induced on the large orbit of N contains a Hamilton cycle with at least one double edge inthe corresponding multigraph or two components each of which contains a Hamilton cycle withdouble edges in the corresponding multigraph. For this purpose we now proceed with the analysisof the structure of basic orbital graphs (and corresponding suborbits) arising from the action ofPSL(2 , q ) on cosets of PGL(2 , q ) given in Row 5 of Table 3. We apply the approach taken in[61] where the computation of suborbits is done using the fact that PSL(2 , q ) ∼ = P Ω − (4 , q ) andthat the action of PSL(2 , q ) on the cosets of PGL(2 , q ) is equivalent to the induced action of P Ω − (4 , q ) on nonsingular 1-dimensional vector subspaces. For the sake of completeness, we givea more detailed description of this action together with a short explanation of the isomorphismPSL(2 , q ) ∼ = P Ω − (4 , q ) (see [34, p.45] for details).Let F q = F q ( α ), where α = θ for F ∗ q = h θ i , and let φ ∈ Aut ( F q ) be the Frobeniusautomorphism of F q defined by the rule φ ( a ) = a q , a ∈ F q . (Note that φ is an involution.) Let W = h w , w i = F q be a natural SL(2 , q )-module. Then SL(2 , q ) acts on W in a natural way.24n particular, the action of g = (cid:20) a bc d (cid:21) ∈ SL(2 , q ) on W is given by w g = a w + c w , w g = c w + d w . Let W be an SL(2 , q )-module with the underlying space W and the action of SL(2 , q ) definedby the rule w ∗ g = w g φ , where g = ( a ij ) ∈ SL(2 , g ) and g φ = ( φ ( a ij ) ij ) = ( a qij ). One can nowsee that · : W ⊗ W × SL(2 , q ) → W ⊗ W defined by the rule( w ⊗ w ′ ) · g = w g ⊗ w ′ ∗ g = w g ⊗ w ′ g φ is an action of SL(2 , q ) on the 4-dimensional space W ⊗ W (that is, on a tensor product of W and W ). The kernal of this action equals Z (SL(2 , q )), and thus this is in fact a 4-dimensionalrepresentation of G = PSL(2 , q ) (an embedding of G into GL(4 , q )). Further, the set B = { v , v , v , v } , where v = w ⊗ w , v = w ⊗ w , v = w ⊗ w + w ⊗ w , v = α ( w ⊗ w − w ⊗ w ) , is a basis for W ⊗ W over F q .Since G fixes the 4-dimensional space V = span F q ( B ) over F q it can be viewed as a subgroup ofGL(4 , q ). A non-degenerate symplectic form f of W and W defined by f ( w , w ) = − f ( w , w ) =1 and f ( w , w ) = f ( w , w ) = 0 is fixed by SL(2 , q ). It follows that G fixes a non-degeneratesymmetric bilinear form of W ⊗ W defined by the rule( w ′ ⊗ w ′ , w ′′ ⊗ w ′′ ) = f ( w ′ , w ′′ ) f ( w ′ , w ′′ ) . Then we have (( v i , v j )) × = − θ , and so for x = P i =1 x i v i ∈ V and y = P i =1 y i v i ∈ V the symmetric form ( x , y ) and the associatedquadratic form Q are given by the rules( x , y ) = x y + x y − x y + 2 θx y and Q ( x ) = 12 ( x , x ) = x x − x + θx . By computation it follows that Q has q + 1 singular 1-dimensional subspaces of V . As for theremaining q ( q +1) nonsingular 1-dimensional subspaces, G has two orbits {h v i (cid:12)(cid:12) Q ( v ) = 1 , v ∈ V } and {h v i (cid:12)(cid:12) Q ( v ) ∈ F ∗ q \ S ∗ , v ∈ V } which are interchanged by a diagonal automorphism of G . LetΩ be the first of these two orbits. Then the action of G on H is equivalent to the action of G onΩ. By comparing their orders, we get PSL(2 , q ) ∼ = P Ω − (4 , q ). The following result characterizingsuborbits of the action of G on the cosets of PGL(2 , q ) in the context of the action of P Ω − (4 , q )on Ω was proved in [61]. 25 roposition 6.3 [61, Lemma 4.1] For any h v i ∈ Ω , the nontrivial suborbits of the action of G on Ω (that is, the orbits of G h v i ) are the sets S ± λ = {h x i ∈ Ω (cid:12)(cid:12) ( x , v ) = ± λ } , where λ ∈ F q , and(i) |S | = q ( q ∓ for q ≡ ± (ii) |S ± | = q − (iii) |S ± λ | = q ( q + 1) with λ − ∈ N ∗ ; (iv) |S ± λ | = q ( q − with λ − ∈ S ∗ .Moreover, all the suborbits are self-paired. Let X = X ( G, H, S λ ) be the basic orbital graph associated with S λ , and take ρ = (cid:20) (cid:21) ∈ G. (For simplicity reasons we refer to the elements of G as matrices; this should cause no confusion.)Clearly, ρ is of order q . For k ∈ F q we have v ρ k = v , v ρ k = k v + v + k v , v ρ k = 2 k v + v , v ρ k = v , and so ρ k maps the vector x = P i =1 x i v i ∈ V to x ρ k = ( x + k x + 2 kx ) v + x v + ( kx + x ) v + x v . Identifying x with ( x , x , x , x ) we have x ρ k = ( x + k x + 2 kx , x , kx + x , x ) . One can check that for k = 0 we have h x ρ k i 6 = h x i , and thus ρ is ( p, q )-semiregular. Let P = h ρ i ,and let P be the set of orbits of P . These orbits will be referred to as blocks. The set Ω decomposesinto two subsets each of which is a union of blocks from P : I = h (0 , , x , x ) i P = {h (2 kx , , x , x ) i (cid:12)(cid:12) k ∈ F q } , where − x + θx = 1. L = h ( x , x , , x ) i P = {h ( x + k x , x , kx , x ) i (cid:12)(cid:12) k ∈ F q } , where x = 0 and x x + θx = 1.Note that the subset I contains q ( q +1)2 vertices which form q +12 blocks, and the subset L contains q ( q − vertices which form q ( q − blocks. By I P and L P , we denote, respectively, the set of blocksin I and L ; that is, P = I P ∪ L P . Remark 6.4
Note that N = N G ( P ) = h (cid:20) a b a − (cid:21) (cid:12)(cid:12) a ∈ h α i , b ∈ F q i , where h α i denotes the multiplicative group generated by α . One may check directly that I P consists precisely of the orbits of N of length q and that L is the orbit of N of length q ( q − .26n the next lemma we observe that X hLi and X hLi P are vertex-transitive and show that thebipartite subgraph of X P induced by I P and L P is a complete bipartite graph. Lemma 6.5
With the above notation, the following hold:(i) The induced subgraph X hLi and the quotient graph X hLi P are both vertex-transitive.(ii) For h x i P ∈ I P and h y i P ∈ L P we have d ( h x i P, h y i P ) = (cid:26) , λ = 02 , λ = 0 . Proof.
By Lemma 6.2, N is transitive on L , and so the induced subgraph X hLi and the quotientgraph X hLi P are both vertex transitive, and thus (i) holds.To prove (ii), take two arbitrary vertices h x i = h (0 , , x , x ) i ∈ I and h y i = h ( y , y , , y ) i ∈L . Then y = 0 and x = 0, and h x i ∼ h y ρ k i if and only if( x , y ρ k ) = ((0 , , x , x ) , ( y + k y , y , ky , y )) = ± λ, that is, if and only if − x ky + 2 θx y = ± λ. (14)From (14) we get that k = θx y ∓ λx y and so for given h x i and h y i we have a unique solution for k if λ = 0 and two solutions if λ = 0. It follows that for h x i P ∈ I P and h y i P ∈ L P we have d ( h x i P, h y i P ) = 1 or 2, depending on whether λ = 0 or λ = 0, completing part (ii) of Lemma 6.5.In what follows, we divide the proof into two cases depending on whether λ = 0 or λ = 0. S Let ε = (cid:26) , q ≡ , , q ≡ , . The following lemma gives us the number of edges inside a block and between two blocks from L P for the orbital graph X ( G, H, S ). Lemma 6.6
Let X = X ( G, H, S ) . Then for h x i ∈ L the following hold:(i) d ( h x i P ) = ε ,(ii) d ( h x i P, h y i P ) = 1 for q +12 blocks h y i P ∈ L P ,(iii) d ( h x i P, h y i P ) = 2 for ( q − q − ε + 1)) blocks h y i P ∈ L P if q ≡ , and for ( q − q − ε + 1)) blocks h y i P ∈ L P if q ≡ . Proof.
Fix a vertex h x i = h (1 , , , i ∈ L . For any h y i = h ( y , y , , y ) i ∈ L , where y = 0, wehave h x i ∼ h y i ρ k if and only if y + ( k + 1) y = 0, and therefore, since y y + θy = 1, if andonly if k = − y − + θ ( y − y ) − . (15)27t follows from (15) that h x i is adjacent to one vertex in the block h y i P ∈ L P if k = 0 and to twovertices in this block if k = 0. Clearly, k = 0 if and only if θy = 1 + y . (16)Proposition 2.5 implies that (16) has q + 1 solutions for ( y , y ), and therefore since h y i = h− y i wehave a total of q +12 choices for h y i . This implies that d ( h x i P, h y i P ) = 1 for q +12 blocks h y i P ∈ L P ,proving part (ii).To prove part (i), take y = ± x = ± (1 , , , h x i P if and only if k = −
2. This equation has solutions if and only if q ≡ , X hh x i P i is a q -cycle for q ≡ , qK if q ≡ , . Finally, to prove part (iii) let m be the number of blocks h y i P ∈ L P for which d ( h x i P, h y i P ) =2. Suppose first that q ≡ X is of valency q ( q − d ( h x i P ) = ε and that h x i is adjacent to ( q + 1) vertices in the set I and to exactlyone vertex from q +12 blocks in L P , we have m = 12 ( 12 q ( q − − q + 12 − q + 12 − ε ) = 14 ( q − q − ε )) . Suppose now that q ≡ X in the above computationwith q ( q + 1) we obtain, as desired, that m = 14 ( q − q − ε )) . We are now ready to prove existence of a Hamilton cycle in X ( G, H, S ). Proposition 6.7
The graph X = X ( G, H, S ) is hamiltonian. Proof.
Let X hLi ′ be the graph obtained from X hLi by deleting the edges between any twoblocks B , B ∈ L P for which d ( B , B ) = 1 (see Lemma 6.6(ii)). By Lemma 6.5, X hLi P isvertex-transitive, and consequently one can see that also X hLi ′P is vertex-transitive.If q ≡ X hLi ′P is of valency m = ( q − q − ε )). If, however, q ≡ X hLi ′P is of valency m = ( q − q − ε )).If q = 5 then ε = 0 and m = ( q − q − ε )) = 2. If q ≥ q − q − ε ) ≥ q ≡ q − q − ε ) ≥ q ≡ m = 14 ( q − (2 ± q − ε )) ≥ q ( q − |L P | . Suppose first that X hLi ′P is connected. If q = 5, then X hLi ′P is just a cycle C . For q ≥
7, byProposition 2.10, X hLi ′P admits a Hamilton cycle, say C again. Clearly C is also a Hamilton cycleof X hLi P . Form C a Hamilton cycle in X P can be constructed by choosing arbitrarily ( q + 1) / q + 1) / N in X P . By Lemma 4.6, this lifts to a Hamilton cycle in X .Next, suppose that X hLi ′P is disconnected. For q = 5, since X hLi ′P is a vertex transitive graphof order 10, it must be a union of two 5-cycles. For q ≥
7, since m ≥ |L P | , it follows that X hLi ′P U = U U · · · U l , and U ′ = U ′ U ′ · · · , U ′ l , where l = q ( q − . Choose any two isolated vertices W and W and construct the cycle D = W U W U ′ W .Choose arbitrarily ( q + 1) / − U ∪ U ′ and replace them by 2-paths having as centralvertices the remaining ( q + 1) / − X P , which,by Lemma 4.6, lifts to a Hamilton cycle in X . S λ with λ = 0 Proposition 6.8
The graph X = X ( G, H, S ± λ ) , where λ = 0 , is hamiltonian. Proof.
As in the proof of Lemma 6.6, fix a vertex h x i = h (1 , , , i ∈ L . For any h y i = h ( y , y , , y ) i ∈ L , y = 0, we have y ρ k = ( y + k y , y , ky , y ), and so h x i ∼ h y ρ k i if and onlyif y + ( k + 1) y = ± λ , which implies, since y y + θy = 1, that k = ± λy − − y − + θ ( y − y ) − . It follows that there are at most four solutions for k . Hence each vertex in L is adjacent to atmost four vertices in the same block from L P (including the block containing this vertex).Let m be the valency of X hLi P . Since, by Proposition 6.3, the valency of X is, respectively, q − q − q and q + q , we get that m ≥ |L P | = q ( q − provided m ≥
14 (( q − j ) − ( q + 1) − ≥
14 ( q − q − j − ≥ q ( q − , where j ∈ { , q, − q } for q ≥ j ∈ { , − q } for q = 5. One can check that this inequality holdsfor all q ≥
5. We can therefore conclude that X hLi P , which is vertex-transitive by Lemma 6.5,has at most two connected components. The rest of the argument follows word by word from theargument given in the proof of Proposition 6.7, since, by Lemma 6.5, d ( h x i P, h y i P ) = 2, for any h x i P ∈ I P and h y i P ∈ L P . PSL(2 , p ) In this section the existence of Hamilton cycles in basic orbital graphs arising from the groupaction in Row 6 of Table 3 is considered. Let us remark that subdegrees of all primitive permu-tation representations of PSL(2 , k ) were calculated in [64]. This thesis is quite unavailable, butsome extractions appeared in [22].Observe first that in order for q = ( p + 1) / p ≡ p ≥
13. Throughout this section let F = F p be a finitefield of order p , and let F ∗ , S ∗ and N ∗ be defined as in Subsection 2.5, that is: F ∗ = F \ { } , S ∗ = { s : s ∈ F ∗ } and N ∗ = F ∗ \ S ∗ .In the description of the graphs arising from the action of G = PSL(2 , p ) on the set H of rightcosets of H = D p − we follow [55]. (For further details as well as all the proofs see [55].) For29implicity reasons we refer to the elements of G as matrices; this should cause no confusion. Wemay choose H to consist of all the matrices of the form (cid:20) x x − (cid:21) ( x ∈ F ∗ ) and (cid:20) − xx − (cid:21) ( x ∈ F ∗ ) . Note that, since p ≥ H is a dihedral subgroup D p − . Further, let g = (cid:20) a bc d (cid:21) ∈ G be fixed. Then each element of the right coset Hg is either of the form (cid:20) ax bxcx − dx − (cid:21) or of the form (cid:20) cx dx − ax − − bx − (cid:21) ( x ∈ F ∗ ) . Moreover, a typical element of the left coset gH is either of the form (cid:20) ax bx − cx dx − (cid:21) or of the form (cid:20) bx − − axdx − − cx (cid:21) ( x ∈ F ∗ ) . The computation and description of the suborbits of G acting by right multiplication on the set H of the right cosets of H in G depends heavily on the concise description of the elements of H .If g satisfies ab = 0, define ξ ( g ) = ad and η ( g ) = a − b , and call χ ( g ) = ( ξ ( g ) , η ( g )) the character of g . The following proposition, proved in [55, Lemmas 2.1 and 2.2], recalls basic properties ofcharacters. Proposition 7.1 [55, Lemmas 2.1 and 2.2]
Let χ ( g ) = ( ξ, η ) .(i) If abcd = 0 and g ′ ∈ Hg then either χ ( g ′ ) = ( ξ, η ) or χ ( g ′ ) = (1 − ξ, ξη/ ( ξ − .(ii) If ab = 0 and g ′ ∈ gH then either χ ( g ′ ) = ( ξ, yη ) or χ ( g ′ ) = (1 − ξ, − yη − ) for some y ∈ S ∗ . Let ≈ be the equivalence relation on F × F ∗ defined by( ξ, η ) ≈ (1 − ξ, ξηξ − ξ = 0 , . (17)There is then a natural identification of the sets H and ( F × F ∗ ) / ≈ ∪ {∞} where ∞ corresponds to H and ( ξ, η ) corresponds to the coset Hg satisfying χ ( g ) = ( ξ, η ). This identification will be usedthroughout the rest of this section. Note that the identification is direct with ( F ∗ \ { } × F ∗ ) / ≈ ∪{∞} ∪ { , } × F ∗ . The symbol ∞ corresponds to the subgroup H and { , } × F ∗ representsall those cosets which contain at least one matrix with exactly one of the entries equal to zero.In summary, all matrices which have two entries equal to zero are in the subgroup H , and allmatrices with exactly one of the entries equal to zero belong to 2( p −
1) cosets of H with typicalrepresentatives (cid:20) y (cid:21) and (cid:20) y − y − (cid:21) ( y ∈ F ∗ )with respective characters (1 , y ) and (0 , y ). Finally, all remaining cosets in H contain matriceswith no entry equal to zero, where we have to bring in the equivalence relation ≈ on charactersdefined by (17). 30or each ξ ∈ F define the following subsets of H (in fact subsets of ( F × F ∗ ) / ≈ ∪{∞} if ξ = 0 , H throughout this section, which should causeno confusion: S + ξ = { ( ξ, η ) : η ∈ S ∗ } , S − ξ = { ( ξ, η ) : η ∈ N ∗ } , S ξ = S + ξ ∪ S − ξ . Observe that (because of the equivalence relation ≈ ) the sets {S + ξ , S − ξ } and {S +1 − ξ , S − − ξ } coincidefor ξ = 0 ,
1. Moreover, since ( , η ) ≈ ( , − η ), it follows that the cardinality of S ξ is p − ξ = when the cardinality is p − . Similarly, the cardinalities of S + ξ and S − ξ are p − exceptfor ξ = when the cardinalities are p − . The following result proved in [55] determines all thesuborbits of the action of G on H . The suborbits given in the theorem are summarized in Table 4. Theorem 7.2 [55, Theorem]
The action of G on H has(i) p +74 suborbits of length p − , all of them self-paired. These are S +0 ∪ S +1 , S − ∪ S − and S ξ for all those ξ = which satisfy ξ − − ∈ N ∗ .(ii) p − suborbits of length p − , namely S + ξ and S − ξ , where ξ = and ξ − − ∈ S ∗ . Amongthem the self-paired suborbits correspond to all those ξ for which both ξ and ξ − belong to S ∗ and so their number is p − if p ≡ and p − if p ≡ .(iii) 2 suborbits of length p − , namely S + and S − which are self-paired if and only if p ≡ . Example 7.3
The smallest admissible pair of primes p = 13 and q = 7 gives rise to the action of G = PSL(2 ,
13) on cosets of H = D with the following suborbits:(i) S +0 ∪ S +1 , S − ∪ S − , S , S , S of size 12, all of them self-paired,(ii) S +4 , S − of size 6, all of them self-paired,(iii) S +6 , S − of size 6, which are not self-paired, and(iv) S +7 , S − of size 3 which are not self-paired.Therefore each of the corresponding generalized orbital graphs is a union of the graphs X ( G, H, W )with W ∈ {S +0 ∪ S +1 , S − ∪ S − , S , S , S +4 , S − , S , S , S } .The following proposition follows from [61]. Consequently, only seven different types of basicorbital graphs arising from the action of PSL(2 , p ) on cosets of D p − need to be considered. Proposition 7.4 [61, Table VI]
The basic orbital graphs arising from Rows 6 and 7 of Table 4 areisomorphic, that is, X ( G, H, S + ) ∼ = X ( G, H, S − ) . Also, the basic orbital graphs arising from Rows8 and 9 of Table 4 are isomorphic, that is, X ( G, H, S + ξ ) ∼ = X ( G, H, S − ξ ) , where ξ ∈ S ∗ ∩ S ∗ + 1 and ξ = , . W Conditions on ξ val ( X ) Conditions on p d ( V ∞ ) d ( V ∞ , V x ) x ∈ F ∗ S ξ ξ = , p − ξ ∈ S ∗ ∩ N ∗ + 1 or ξ ∈ N ∗ ∩ S ∗ + 12 S ξ ξ = p − ξ ∈ N ∗ ∩ N ∗ + 13 S +0 ∪ S +1 p − p − / x ∈ S ∗ x ∈ N ∗ S − ∪ S − p − p − / x ∈ S ∗ x ∈ N ∗ S ξ = ( p − / p ≡ S + ξ = ( p − / p ≡ x ∈ S ∗ x ∈ N ∗ S − ξ = ( p − / p ≡ x ∈ S ∗ x ∈ N ∗ S + ξ ξ = , p − / x ∈ S ∗ ξ ∈ S ∗ ∩ S ∗ + 1 0 if x ∈ N ∗ S − ξ ξ = , p − / x ∈ S ∗ ξ ∈ S ∗ ∩ S ∗ + 1 2 if x ∈ N ∗ Table 4:
The list of all basic orbital graphs X = X ( G, H, W ), where W is a self-paired union of suborbits describedin Theorem 7.2. In the last two columns valencies of d ( V ∞ ) and d ( V ∞ , V x ), x ∈ F ∗ , are listed. By Proposition 7.4graphs arising from suborbits in Rows 6 and 7 are pairwise isomorphic. Also, graphs arising from suborbits in Rows8 and 9 are pairwise isomorphic. With the explicit description of the suborbits of G on H the construction of the correspondinggeneralized orbital graphs X = X ( G, H, W ), where W is a self-paired union of suborbits of G , isrelatively simple. Namely, the edge set of X is precisely the set {{ Hg, Hwg } : g ∈ G, w ∈ W} .The description of these graphs X = X ( G, H, W ), where W is a self-paired union of suborbitsof G is best done via a ‘factorization modulo’ the Sylow p -subgroup P = h (cid:20) (cid:21) i = { (cid:20) a (cid:21) : a ∈ F } . Observe that P has ( p + 1) / H . These are V ∞ = { H (cid:20) a (cid:21) : a ∈ F } and V x = { H (cid:20) x − x − (cid:21) (cid:20) a (cid:21) : a ∈ F } = V − x , x ∈ F ∗ .
32n the proofs below we will often use the fact that V x , x ∈ F ∗ , contains both H (cid:20) x − x − (cid:21) and H (cid:20) − xx − (cid:21) . Note also that, using the above mentioned identification, we have V ∞ = {∞ , (1 , , (1 , , . . . , (1 , p − } and V x = { (0 , x ) , (0 , − x ) } ∪ { ( ξ, x ) , ( ξ, − x ) : ξ ∈ F ∗ \ { }} , x ∈ F ∗ . The generator ρ = (cid:20) (cid:21) ∈ P is a (( p + 1) / , p )-semiregular automorphism of X . Let P = { V ∞ } ∪ { V x : x ∈ F ∗ } be the set oforbits of ρ , and consider the corresponding quotient graph X P and quotient multigraph X ρ (seeSubsection 2.3). The following proposition gives the values of d ( V ∞ ) and d ( V ∞ , V x ), x ∈ F ∗ , forthe basic orbital graphs X ( G, H, W ) (see Theorem 7.2). Proposition 7.5
The valencies d ( V ∞ ) and d ( V ∞ , V x ) , x ∈ F ∗ , for a basic orbital graph X = X ( G, H, W ) are as given in Table 4. Proof.
Note that H ∈ V ∞ is adjacent to all the vertices of the form Hw , where w ∈ W , whichbelong to V ∞ if and only if its character is of the form (1 , η ), η ∈ F ∗ . Since, by Theorem 7.2, S +0 ∪ S +1 and S − ∪ S − are the only two suborbits with nontrivial intersection with S we obtainthe value of d ( V ∞ ) as given in Table 4.To determine the values for d ( V ∞ , V x ) let us consider the neighbors of H . Note that a repre-sentative of a coset Hg outside V ∞ adjacent to H is of the form (cid:20) z y − z y (cid:21) for a suitable z ∈ F ∗ . Now, if this neighbor is inside the orbit V x then there exists j ∈ F such that (cid:20) z y − z y (cid:21) = (cid:20) x − x − (cid:21) (cid:20) j (cid:21) = (cid:20) j + x − x − − jx (cid:21) . Hence, recalling the equivalence relation (17) used for identification of cosets and characters, either( y, z ) or (1 − y, yzy − ) is equal to ( − jx , j + x ). Now, the two equations ( y, z ) = ( − jx , j + x ) and(1 − y, yzy − ) = ( − jx , j + x ) give solutions j = yzy − , x = z − y and j = z, x = zy − − z − y . Since V x = V − x and yzy − = y if y = we can conclude that for W ∈ {S +0 ∪S +1 , S − ∪S − , S ξ , S + ξ , S − ξ } either V ∞ and V x are adjacent in X ρ with a double edge or there is no edge between them, whereasfor W ∈ {S , S + , S − } all the edges in X ρ containing the vertex V ∞ are simple edges. Applyingthe conditions for suborbits given in Theorem 7.2 one can now obtain the values of d ( V ∞ , V x ) forall possible basic suborbits W . 33 quasi-semiregular action is a natural generalization of a semiregular action (see Subsec-tion 2.3). Following [35], we say that a group G acts quasi-semiregularly on a set V if thereexists an element v in V such that v is fixed by any element of G , and G acts semiregularly on V \ { v } . If G is nontrivial, then v is uniquely determined, and is referred to as the fixed point of G . A nontrivial automorphism g of a graph X is called quasi-semiregular if the group h g i actsquasi-semiregularly on V ( X ). Equivalently, g fixes a vertex and the only power g i fixing anothervertex is the identity mapping. If a group G is quasi-semiregular on the vertex set of the graphwith m + 1 orbits, then the graph is called a quasi m -Cayley graph on G . If G is cyclic andquasi-semiregular with two nontrivial orbits then the graph is said to be a quasi-bicirculant .In the next proposition we prove that the quotient graph X P of a generalized orbital graph X is a quasi-bicirculant. The corresponding group automorphism is given by a diagonal matrixwhose diagonal consists of an appropriate pair of generators of S ∗ . More precisely, let σ = (cid:20) z z − (cid:21) , where h z i = S ∗ . (18) Proposition 7.6
Let X = X ( G, H, W ) . Then X P is a quasi-bicirculant. The correspondingquasi-semiregular action on X P is given by the subgroup generated by σ . Moreover, the fixedpoint of this action is V ∞ and the two nontrivial orbits are O ( S ∗ ) = { V x : x ∈ S ∗ } and O ( N ∗ ) = { V x : x ∈ N ∗ } . Proof.
Note first that σ ∈ H . Observe that for an element (cid:20) a (cid:21) ∈ P , a ∈ F , we have σ − (cid:20) a (cid:21) σ = (cid:20) az − (cid:21) ∈ P, and consequently h σ i ≤ N G ( P ). It follows that for every a ∈ F we have H (cid:20) a (cid:21) σ = Hσ (cid:20) az − (cid:21) = H (cid:20) az − (cid:21) ∈ V ∞ . We can conclude that V ∞ is fixed by h σ i . Now let us look at the action of σ on elements from anorbit V x , x ∈ F ∗ . Recall that elements of V x are of the form H (cid:20) x − x − (cid:21) (cid:20) a (cid:21) , where a ∈ F .
Applying the action of σ on any of these elements gives H (cid:20) x − x − (cid:21) (cid:20) a (cid:21) σ = H (cid:20) x − x − (cid:21) σ (cid:20) az − (cid:21) = H (cid:20) z xz − − x − z (cid:21) (cid:20) az − (cid:21) ∈ V xz . It follows that h σ i cyclically permutes the orbits of P , and we can represent σ as a permutationof the vertex set of X P in the following way σ ( V ∞ ) = V ∞ and σ ( V x ) = V xz . z ∈ S ∗ it is now clear that h σ i is a quasi-semiregular group of automorphisms of X P (aswell as of X ρ ) with one of the nontrivial orbits consisting of V x , x ∈ S ∗ , and the other consisting V x , x ∈ N ∗ .In subsequent lemmas and propositions the following observations on characters of adjacentvertices in X P will be frequently used. Let X = X ( G, H, W ), where W is one of the basic self-paired union of suborbits given in Rows 1, 2, 5, 6, 8 and 9 of Table 4, and let V y ∈ V ( X ρ ), y ∈ F ∗ .Then a representative of a coset adjacent to a coset in V y is of the form " η ξ − η ξ y − y − (cid:21) = " − ηy − y ξ − η − ξy − y ( ξ − η , for some η ∈ F ∗ , where ξ determines the suborbit W (see Table 4). If this neighbor is inside an orbit V x , x ∈ F ∗ ,then there exists j ∈ F such that " − ηy − y ξ − η − ξy − y ( ξ − η ≡ (cid:20) x − x − (cid:21) (cid:20) j (cid:21) = (cid:20) j + x − x − − jx − (cid:21) . Hence, in view of the equivalence relation (17), one can see that either( ( ξ − y − η ) η , y y − η ) = ( − jx , j + x ) or ( ( ξ − y − η ) η , y y − η ) = (1 + jx , j ) . (19)This gives us that j , = 12 ( y − x ± p ( x + y ) − xyξ ) , j , = 12 ( y − x ± p ( x − y ) + 4 xyξ ) , (20)and that η , = y − y y + x ± p ( x + y ) − xyξ , η , = y − y y + x ± p ( x − y ) + 4 xyξ , (21)Whenever we need to compare the values of η i for different pairs of orbits V x , V y and V z , V w we will write η i ( x, y ) and η i ( z, w ). Proposition 7.7
Let X = X ( G, H, W ) , where W is one of the basic self-paired union of suborbitsgiven in Rows 1, 2, 5, 6, 7, 8 and 9 of Table 4. Then for y ∈ F ∗ we have d ( V y ) = , ξ ∈ N ∗ ∩ N ∗ + 12 , ξ ∈ S ∗ ∩ N ∗ + 1 or ξ ∈ N ∗ ∩ S ∗ + 14 , ξ ∈ S ∗ ∩ S ∗ + 10 , ξ = 1 / and p ≡ , ξ = 1 / and p ≡ , where ξ indicates the subscript ξ at S ǫξ , ǫ = ± , in self-paired union W , as given in Table 4. Proof.
Observe that d ( V y ) is given by the number of solutions of the equations given in (20)and (21) for x = y .In the next five subsections we prove the existence of Hamilton cycles for the basic orbitalgraphs arising from group actions given in Table 4.35 .1 Case S ξ with ξ = , (Rows 1 and 2 of Table 4) Proposition 7.8
Let X = X ( G, H, W ) , where W is one of the basic self-paired unions of suborbitsgiven in Rows 1 and 2 of Table 4. Then X is hamiltonian. Proof.
Observe first that Proposition 7.5 implies that d ( V ∞ ) = ( p − / d ( V ∞ , V x ) = 2 forevery x ∈ F ∗ . Note also that d ( V x , V y ) ≤ x, y ∈ F ∗ , and that, by Proposition 7.7, d ( V x ) ≤ x ∈ F ∗ . It follows that each V x is joined to at least ( p − / X P − { V ∞ } .Namely, subtracting from the valency p − X the valency d ( V x , V y ) and the maximal possiblevalency d ( V x ) ≤ V x we are left with at least p − X ρ incident with V x .But p ≡ X P − { V ∞ } is of valency at least ( p − / X P − { V ∞ } is regular then, since p ≥
13, we apply Proposition 2.10 to get a Hamiltoncycle in X P − { V ∞ } . Since d ( V ∞ , V x ) = 2 for every x ∈ F ∗ , we can extend this Hamilton cyclein X P − { V ∞ } to a Hamilton cycle in X P , and then apply Lemma 4.6 to conclude that X ishamiltonian.We may therefore assume that X P − { V ∞ } is not regular. Without loss of generality we mayassume that val X P ( V x ) > val X P ( V y ) for x ∈ S ∗ and y ∈ N ∗ . (Recall that, by Proposition 7.6, thereexists an automorphism of X ρ which cyclically permutes vertices in the set O ( S ∗ ) = { V x : x ∈ S ∗ } and vertices in the set O ( N ∗ ) = { V y : y ∈ N ∗ } . Consequently, vertices inside each of these twosets are of the same valency.)Since we are in the case S ξ = S + ξ ∪S − ξ the solutions of (20) and (21) for x = y depends solely on ξ . Consequently, d ( V x ) = d ( V y ) for all x, y ∈ F ∗ . Suppose first that d ( V x ) = 0, x ∈ F ∗ . Note thatthis happens in Row 2 of Table 4. Combining together the additional facts that d ( V ∞ , V x ) = 2for every x ∈ F ∗ , that p ≡ V x , x ∈ F ∗ , in X P satisfies val X P ( V x ) ≥ (cid:26) ( p + 3) / , x ∈ N ∗ ( p + 7) / , x ∈ S ∗ , and so the existence of a Hamilton cycle in X P follows by Proposition 2.9. Namely, the conditionsof Proposition 2.9 are vacuously satisfied since S i = ∅ for every i < | V ( X P ) | /
2. Clearly thisHamilton cycle contains at least one double edge in X ρ (for example, all edges incident with V ∞ are such double edges), Lemma 4.6 implies that X is hamiltonian.In the remaining case of Row 1 of Table 4 we have, in view of (20), that d ( V x ) = 2, x ∈ F ∗ .Then using the same arguments as in the previous case, we may assume that the valency of avertex V x , x ∈ F ∗ , in X P satisfies val X P ( V x ) ≥ (cid:26) ( p − / , x ∈ N ∗ ( p + 3) / , x ∈ S ∗ , and so the existence of a Hamilton cycle in X P now follows by Proposition 2.9. Namely, i =( p − / i < | V ( X P ) | / p + 1) / | S ( p − / | 6≤ ( p − / −
1. But | S ( p +1) / − ( p − / − | = | S ( p − / | ≤ ( p − /
4, and so Proposition 2.9 indeed applies. As in the previous paragraph thisHamilton cycle clearly contains at least one double edge in X ρ and so Lemma 4.6 implies that X is hamiltonian. 36 .2 Cases S +0 ∪ S +1 and S − ∪ S − (Rows 3 and 4 of Table 4) Proposition 7.9
Let X = X ( G, H, W ) , where W ∈ {S +0 ∪ S +1 , S − ∪ S − ) , be one of the graphsarising from Rows 3 or 4 of Table 4. Then X is hamiltonian. Proof.
Suppose that W = S +0 ∪ S +1 . By Proposition 7.5 we have d ( V ∞ ) = ( p − / d ( V ∞ , V x ) = (cid:26) , x ∈ S ∗ , x ∈ N ∗ . We now need to compute the valency val X P −{ V ∞ } ( V y ) of V y in the graph X P − { V ∞ } . Thecharacter of a coset in S +0 ∪ S +1 is either of the form (0 , η ) or (1 , η ), η ∈ S ∗ , with respectiverepresentatives E = (cid:20) η − η − (cid:21) and E = (cid:20) η (cid:21) . Then a representative of a coset adjacent to a coset in V y is either of the form E · (cid:20) y − y − (cid:21) or of the form E · (cid:20) y − y − (cid:21) . Further, if this neighbor is inside an orbit V x , x ∈ F ∗ , then there exists j ∈ F such that either E · (cid:20) y − y − (cid:21) ≡ (cid:20) x − x − (cid:21) (cid:20) j (cid:21) or E · (cid:20) y − y − (cid:21) ≡ (cid:20) x − x − (cid:21) (cid:20) j (cid:21) . With the equivalence relation (17) in mind, one can see that the four cases given in Table 5 arise.Let η i be the possible solution for η given in the i -th row of Table 5. Then η η = η η = y . Thisimplies that either ( η , η ) ∈ S ∗ × S ∗ or ( η , η ) ∈ N ∗ × N ∗ , and that either ( η , η ) ∈ S ∗ × S ∗ or( η , η ) ∈ N ∗ × N ∗ . Row j η y − x xy/ ( x − y )2 y xy/ ( x + y )3 − x y ( x + y ) /x y ( x − y ) /x Table 5: Conditions on j and η for existence of an edge between V x and V y , x, y ∈ F ∗ , in X = X ( G, H, W ), where W = S +0 ∪ S +1 .For x = y Rows 1 and 4 of Table 5 give no solution. Hence η = x/ η = 2 x are the onlysolutions, and thus for x ∈ S ∗ we have d ( V x ) = 2 ⇔ ∈ S ∗ ⇔ p ≡ , and for x ∈ N ∗ we have d ( V x ) = 2 ⇔ ∈ N ∗ ⇔ p ≡ .
37e now compute the valencies of the vertices in the quotient graph X P . Since X P is a quasi-bicirculant, in order to compute the valency of an arbitrary vertex V x ∈ O ( S ∗ ) it suffices tocompute the valency val X P ( V ) of the vertex V instead of computing the valency of an arbitraryvertex V x ∈ O ( S ∗ ) (in short, we may assume that x = 1). Since W = S +0 ∪ S +1 we must have η ∈ S ∗ . It follows from Table 5 that if V y ∈ O ( S ∗ ) is adjacent to V in X P then d ( V , V y ) ∈ { , } ,and so either y ∈ S ∗ ∩ S ∗ + 1 or y ∈ S ∗ ∩ S ∗ −
1. Note that for p ≡ ± ∈ ( S ∗ ∩ S ∗ + 1) ∪ ( S ∗ ∩ S ∗ − p ≡ ± | S ∗ ∩ S ∗ + 1 | + 2 − / p − / O ( S ∗ ) adjacent to V if p ≡ | S ∗ ∩ S ∗ +1 | +2) / p +3) / O ( S ∗ ) adjacent to V if p ≡ V y = V − y .) Moreover,( p − / p ≡ p − / O ( S ∗ )adjacent to V when p ≡ val ( V ) X P hO ( S ∗ ) i = (cid:26) ( p − / , p ≡ p + 3) / , p ≡ . Similarly, using Propositions 2.4 and 2.6 for neighbors of V in O ( N ∗ ) we can see that their numberis (( p − / / p + 7) / val ( V ) X P −{ V ∞ } − val ( V ) X P hO ( S ∗ ) i = (cid:26) ( p + 7) / , p ≡ p + 11) / , p ≡ . It remains to calculate the valency of the subgraph X P hO ( N ∗ ) i of X P induced on O ( N ∗ ). Let V x ∈ O ( N ∗ ) be a fixed vertex. If V y ∈ O ( N ∗ ) is adjacent to this vertex V x then we must have x ± y ∈ S ∗ , and thus the number of vertices in O ( N ∗ ) adjacent to V x depends on the cardinalityof the set ( S ∗ ∩ N ∗ + x ) ∪ ( S ∗ ∩ N ∗ − x ) , which is equal to the cardinality of the set ( N ∗ ∩ S ∗ + 1) ∪ ( N ∗ ∩ S ∗ − N ∗ − ∩ S ∗ ) ∪ ( N ∗ + 1 ∩ S ∗ ). Thus, since ± ∈ N ∗ − ∩ S ∗ for p ≡ val ( V x ) X P hO ( N ∗ ) i = (cid:26) ( p + 7) / , p ≡ p − / , p ≡ . If follows that, with the exception of V ∞ which is of valency ( p − /
4, all other vertices in X P are of valency at least ( p + 3) / X P . Now Proposition 2.9implies the existence of a Hamilton cycle in X P . Namely, with the corresponding notation for thesets S i we have | S ( p − / | = 1 ≤ ( p − / −
1. This Hamilton cycle in X ρ clearly has double edges,and so, by Lemma 4.6, it lifts to a Hamilton cycle in X .The hamiltonicity of the graph X for W = S − ∪ S − is determined in an analogous way. Weomit the details. S (Row 5 of Table 4) roposition 7.10 Let p ≡ and let X = X ( G, H, S ) be the graph arising from Row 5of Table 4. Then X is hamiltonian. Proof.
Note that X is of valency ( p − /
2. By Proposition 7.5 we have d ( V ∞ ) = 0 and d ( V ∞ , V y ) = 1 for every y ∈ F ∗ .We now need to compute the valency val X P −{ V ∞ } ( V y ), y ∈ F ∗ . Let x ∈ F ∗ . The number ofedges d ( V y , V x ) between V y and V x in X ρ is obtained from (20) and (21) by letting ξ = 1 /
2. Weobtain j , = j , = 12 ( y − x ± p x + y )and η , = η , = y − y y + x ± p x + y = yx ( ± p x + y − y ) . By Proposition 2.3 it follows that 2 ∈ N ∗ , and so we have d ( V x ) = 0 for every x ∈ F ∗ . Further, if x + y = 0 then y = − x , and so y = ±√− x . Since √− ∈ N ∗ when p ≡ x ∈ S ∗ there exists a unique y ∈ N ∗ such that d ( V x , V y ) = 1, whereas all otheredges in X ρ − { V ∞ } containing V x are double edges. It follows that for every x ∈ F ∗ we have val ( V x ) = ( p − − / p −
54 + 2 = p + 34 . Since val X P ( V ∞ ) = ( p − / X P are of valency more than halfof the order of X P , and so Proposition 2.9 implies the existence of a Hamilton cycle in X P . Sincethis cycle clearly contains a double edge, Lemma 4.6 implies that X is hamiltonian. S + and S − (Rows 6 and 7 of Table 4) In this and the next subsection Hamilton cycles are constructed using the results from Section 3about polynomials of degree 4 that represent quadratic residues at primitive roots.
Proposition 7.11
Let p ≡ and let X = X ( G, H, W ) , where W ∈ { S + , S − } , be one ofthe graphs arising from Row 6 or 7 of Table 4. Then X is hamiltonian. Proof.
By Proposition 7.4 the two graphs are isomorphic, and so we may assume that W = S + .Note that X is of valency ( p − /
4. Since ( p + 1) / p = 17 we may also assumethat p > d ( V ∞ ) = 0 and d ( V ∞ , V x ) = 1 for every x ∈ S ∗ . We now needto compute the valency val X P −{ V ∞ } ( V y ), y ∈ F ∗ . Let x ∈ F ∗ . The number of edges d ( V y , V x )between V y and V x in X ρ is obtained from (20) and (21) by letting ξ = 1 /
2. We obtain j , = j , = 12 ( y − x ± p x + y ) and η , = η , = yx ( ± p x + y − y ) . (22)By Proposition 2.3, 2 ∈ S ∗ , and so d ( V x ) = 2 for every x ∈ F ∗ for which η , = η , = ± x ( √ − ∈ S ∗ . Further, if x + y = 0 then y = − x , and so y = ±√− x and η , = ∓√− x . Since √− ∈ S ∗ for p ≡ x ∈ S ∗ there exits a unique V y ∈ X ρ hO ( S ∗ ) i such that d ( V x , V y ) = 1. All other edges in X ρ − { V ∞ } incident with V x are double edges. Furthermore, alsoall of the edges in X ρ − { V ∞ } incident with V x , x ∈ N ∗ , are double edges.39uppose that x = 1. Then we get from (22) that j , = j , = 12 ( y − ± p y ) and η , = η , = y ( ± p y − y ) . Let us now consider elements of the form 1 + g ∈ F ∗ , where g ∈ F ∗ is a generator of F ∗ , thatis, F ∗ = h g i . Since p ≡ g ∈ F ∗ such that F ∗ = h g i and 1 + g ∈ S ∗ .The element s = g generates S ∗ . We claim that either V is adjacent to V s or V g is adjacentto V sg . This will in turn imply that there is a full cycle either in the induced graph on O ( S ∗ ) or inthe induced graph on O ( N ∗ ). The corresponding values η and η for the pairs V , V s and V g , V gs are, respectively, η , (1 , s ) = s ( ± p s − s ) and η , ( g, gs ) = gs ( ± p s − s ) . Therefore η i ( g, gs ) η i (1 , s ) = g ∈ N ∗ , i ∈ { , } . And consequently, the bicirculant X ρ − { V ∞ } indeed has a full induced cycle C either on the orbit O ( S ∗ ) or on the orbit O ( N ∗ ). More precisely, this cycle is induced by an edge inside one of thetwo orbits of σ and the action of σ on this edge. (Recall that O ( S ∗ ) and O ( N ∗ ) are the two orbitsof the quasi-semiregular automorphism σ from Proposition 7.6).We claim that X ρ − { V ∞ } contains a subgraph isomorphic to a generalized Petersen graph GP (( p − / , k ) for some k ∈ Z ( p − / . In order to prove this we need to show first that the orbitthat does not contain C contains a cycle or a union of cycles induced by the action of σ on an edgein this orbit and second that the bipartite graph between these two orbits contains a matchingpreserved by σ . The latter holds because, by Proposition 7.5, vertex V ∞ is adjacent to all verticesin O ( S ∗ ) and no vertex in O ( N ∗ ), and consequently the connectedness of X implies the existenceof at least one edge with one endvertex in O ( S ∗ ) and one endvertex in O ( N ∗ ). The action of σ onthis edge gives us the desired matching. For the former, there are four possibilities depending onwhether the full cycle is in O ( S ∗ ) or in O ( N ∗ ) and on whether d ( V x ) = 2 for each x ∈ S ∗ or foreach x ∈ N ∗ . A quick analysis based on the valency conditions in the graph X shows that sucha collection of cycles always exists with one exception only. This exception occurs when the fullcycle is in O ( N ∗ ) and d ( V x ) = 2 for x ∈ S ∗ . In this case, however, we can apply Proposition 2.8to see that V is adjacent to V x for some √− = x ∈ S ∗ , and so the corresponding union of cyclesinduced by the action of σ is the collection of cycles we were aiming for. Consequently, X ρ − { V ∞ } contains the desired generalized Petersen graph as a subgraph also in this case. Since p − is even,Proposition 2.11 implies that X ρ − { V ∞ } contains a Hamilton cycle. Of course, this cycle containsan edge of the form V x V y , x, y ∈ S ∗ . Replacing this edge with a path V x V ∞ V y gives a Hamiltoncycle in X ρ . Obviously this Hamilton cycle contains double edges in X ρ , and so, by Lemma 4.6,lifts to a Hamilton cycle in X . S + ξ and S − ξ with ξ = , (Rows 8 and 9 of Table 4) In Proposition 7.15 graphs arising from Rows 8 and 9 of Table 4 are considered. Before statingthis proposition we cover three exceptional cases for which the results about polynomials fromSection 3 cannot be fully applied. The three exceptional pairs ( p, ξ ) are (13 , , , xample 7.12 Let X be a basic orbital graph arising from the action of G = PSL(2 ,
13) onthe cosets of D p − = D from Row 8 or 9 of Table 4. Because of the isomorphism given inProposition 7.4 we may assume that X arises from Row 8 of Table 4, that is, it is associated witha suborbit S + ξ , where ξ ∈ S ∗ ∩ S ∗ + 1, ξ = and ξ = 1.For p = 13 we have( S ∗ ∪ { } ) ∩ ( S ∗ ∪ { } ) + 1 = { , , , } = { , } ∪ { a, a − : a = 4 } . There are two self-paired suborbits of length 6, giving, up to isomorphism, one vertex-transitivegraph of order 13 · S i = { v ji : j ∈ Z } , i ∈ Z , of a (7 , ∅ { , } { , } { , } ∅ ∅ ∅{ , } {± } ∅ ∅ { , } ∅ ∅{ , } ∅ {± } ∅ ∅ { , } ∅{ , } ∅ ∅ {± } ∅ ∅ { , }∅ { , } ∅ ∅ {± } { } { }∅ ∅ { , } ∅ { } {± } { }∅ ∅ ∅ { , } { } { } {± } Note that there is no Hamilton cycle in X ρ (see also Figure 5). There however exists a cycle S S S S S S in X ρ that lifts to a 65-cycle in X containing the edge v v , and there exists a26-cycle containing all the vertices in the orbits S and S and containing the edge v v . Replacingthe edges v v and v v in these two cycles with the edges v v and v v gives a Hamilton cyclein X . S S S S S S S Figure 5:
The orbital graph arising from the action of G = PSL(2 ,
13) on the cosets of D p − = D with respectto a suborbit S + ξ where ξ ∈ S ∗ ∩ S ∗ + 1, ξ = ,
1, given in Frucht’s notation with respect to a (7 , ρ . xample 7.13 Let X be a basic orbital graph arising from the action of G = PSL(2 ,
37) onthe cosets of D p − = D from Row 8 or 9 of Table 4. Because of the isomorphism given inProposition 7.4 we may assume that X arises from Row 8 of Table 4, that is, it is associated witha suborbit S + ξ , where ξ ∈ S ∗ ∩ S ∗ + 1, ξ = and ξ = 1.There are 8 self-paired suborbits of length ( p − / ·
19 = 703 and of valency 18. (For example, this can be checked usingMagma [11].) We may assume that X is one of these graphs.For p = 37 we have( S ∗ ∪ { } ) ∩ ( S ∗ ∪ { } ) + 1 = { , , , , , , , , , } = { , } ∪ { a, a − : a ∈ { , , , }} . The set of primitive roots in F equals R = { , , , , , , , , , , , } . It follows by(20) and (21) that vertices V and V x , x ∈ F ∗ , are adjacent in X P if and only if x + 2(1 − ξ ) x + 1 ∈ S ∗ ∪ { } and η , (1 , x ) = 1 −
21 + x ± p x + 2(1 − ξ ) x + 1 ∈ S ∗ or x − − ξ ) x + 1 ∈ S ∗ ∪ { } and η , (1 , x ) = 1 −
21 + x ± p x − − ξ ) x + 1 ∈ S ∗ . Given an arbitrary τ ∈ R , it follows that V is adjacent to V τ if and only if either τ + 2(1 − ξ ) τ + 1 ∈ S ∗ ∪ { } and η , (1 , τ ) ∈ S ∗ or τ − − ξ ) τ + 1 ∈ S ∗ ∪ { } and η , (1 , τ ) ∈ S ∗ . We apply Proposition 3.7 to conclude that for polynomials f , ( x ) = x ± − ξ ) x + 1 thereexists τ ∈ R such that f j ( τ ) ∈ S ∗ ∪ { } either for j = 1 or for j = 2.Let T + be the subset of R consisting of all those primitive roots τ for which f ( τ ) ∈ S ∗ ∪ { } ,and let T − be the subset of R consisting of all those primitive roots τ for which f ( τ ) ∈ S ∗ ∪ { } .Table 6 gives the list of elements in T + and T − for each ξ . Further, for each element in T + ∪ T − this table also gives information on whether η i (1 , τ ) belongs to S ∗ or not. Checking Table 6 onecan see that for every ξ / ∈ { , } there is at least one τ ∈ R such that V is adjacent to V τ . Weconclude that there is a full cycle in X hO ( S ∗ ) i preserved by the automorphism σ . Note that X P is a quasi-bicirculant with V ∞ as the fixed vertex, and that by Table 4, vertex V ∞ is adjacent onlyto vertices in O ( S ∗ ). Therefore, connectedness of X implies that the bipartite graph induced bythe edges with one endvertex in O ( S ∗ ) and the other in O ( N ∗ ) contains a matching preserved bythe σ . Since, by (21), we have η i ( τ, τ ) η i (1 , τ ) = τ ∈ N ∗ , i ∈ { , , , } , Table 6 also implies that vertices in X hO ( N ∗ ) i are of valency at least 2. Namely, for every ξ thereexist at least one τ ∈ T + ∪ T − such that η i (1 , τ ) / ∈ S ∗ which implies that η i ( τ, τ ) ∈ S ∗ and thus V τ is adjacent to V τ . Therefore, for ξ / ∈ { , } the bicirculant X ρ − { V ∞ } contains a generalizedPetersen graph as a subgraph. Hence, combining together Propositions 2.11 and 2.12 we have42hat X ρ − { V ∞ } contains a Hamilton cycle or at least a Hamilton path with both endvertices in O ( S ∗ ). Since V ∞ is adjacent to all vertices in O ( S ∗ ) we can clearly extend this cycle/path to aHamilton cycle in X P containing at least one double edge in X ρ . By Lemma 4.6 this cycle lifts toa Hamilton cycle in X .We are left with the last two cases ξ ∈ { , } . In both cases Table 6 implies that there is afull cycle in O ( N ∗ ). Namely, η , (1 , τ ) S ∗ implies that η , ( τ, τ ) ∈ S ∗ , and so V τ is adjacentto V τ . Further, for ξ = 10 we have 10 − − ξ ) ·
10 + 1 = 0 and η , (1 ,
10) = 21 ∈ S ∗ , and so V is adjacent to V with a single edge. Similarly, for ξ = 28 we have 11 − − ξ ) ·
11 + 1 = 0and η , (1 ,
11) = 7 ∈ S ∗ , and so V is adjacent to V with a single edge. Since 10 , ∈ S ∗ andtheir squares are not ±
1, we can conclude that vertices in X hO ( S ∗ ) i are of valency at least 2also in these two cases. Therefore, combining together Propositions 2.11 and 2.12, there existsa Hamilton cycle/path in X ρ − { V ∞ } also for ξ ∈ { , } . As before this cycle/path can beextended to a Hamilton cycle in X ρ which, by Lemma 4.6, lifts to a Hamilton cycle in X . ξ τ ∈ T + τ ∈ T + s.t. τ ∈ T − τ ∈ T − s.t. η , (1 , τ ) ∈ S ∗ η , (1 , τ ) ∈ S ∗ ± , ± ± , ± ± , ± , ± , ± ± , ± ± , ± − ± , ± − ± , ± , ± ± ± , ± ± , ± , ± , ± ± , ± − − ± , ± , ± , ± ± , ± ± , ± , ± , ± ± , ± − − ± , ± , ± , ± ± , ± ± , ± , ± , ± ± , ± ± , ± − ± , ± − ± , ± , ± , ± ± , ± ± , ± ± , ± Table 6:
Information about existence of edges in X (PSL(2 , , D , S + ξ ) where ξ ∈ S ∗ ∩ S ∗ + 1, ξ = and ξ = 1. Example 7.14
Let X be a basic orbital graph arising from the action of G = PSL(2 ,
61) onthe cosets of D p − = D from Row 8 or 9 of Table 4. Because of the isomorphism given inProposition 7.4 we may assume that X arises from Row 8 of Table 4, that is, it is associated witha suborbit S + ξ , where ξ ∈ S ∗ ∩ S ∗ + 1, ξ = and ξ = 1. There are 14 self-paired suborbits oflength 30, giving 7 non-isomorphic vertex-transitive graphs of order 61 ·
31 = 1891 and of valency30. (For example, this can be checked using Magma [11].) We may assume that X is one of these7 graphs.For p = 61 we have( S ∗ ∪ { } ) ∩ ( S ∗ ∪ { } ) + 1 = { , , , , , , , , , , , , , , , } = { , } ∪ { a, a − : a ∈ { , , , , , , }} . The set of primitive roots in F equals R = {± , ± , ± , ± , ± , ± , ± , ± } .By Table 4, V ∞ is adjacent to exactly one of the two nontrivial orbits of a quasi (2 , σ . Existence of a Hamilton cycle in each of these graphs can be provedusing the same arguments as in Example 7.13. In particular, with the terminology of Example 7.13one can see from Table 7 that for every ξ there is at least one τ ∈ R such that V is adjacent to V τ . We conclude that there is a full cycle in X hO ( S ∗ ) i preserved by the automorphism σ . Since,by Table 4, vertex V ∞ is adjacent only to vertices in O ( S ∗ ) and since X P is a quasi-bicirculantwith V ∞ as the fixed vertex, we deduce that the bipartite graph induced by the edges with one43ndvertex in O ( S ∗ ) and the other in O ( N ∗ ) contains a matching preserved by the automorphism σ . Next, by (21), we have η i ( τ, τ ) η i (1 , τ ) = τ ∈ N ∗ , i ∈ { , , , } , and so Table 7 implies that vertices in X hO ( N ∗ ) i are of valency at least 2. Namely, for every ξ there exists at least one τ ∈ T + ∪ T − such that η i (1 , τ ) / ∈ S ∗ which implies that η i ( τ, τ ) ∈ S ∗ and thus V τ is adjacent to V τ . We have thus proved that the bicirculant X ρ − { V ∞ } contains ageneralized Petersen graph as a subgraph for every ξ . Hence, combining together Propositions 2.11and 2.12, we have that X ρ − { V ∞ } contains a Hamilton cycle or (at least) a Hamilton path withboth endvertices in O ( S ∗ ). Since V ∞ is adjacent to all of vertices in O ( S ∗ ) this cycle/path canclearly be extend to a Hamilton cycle in X P which contains at least one double edge in X ρ . ByLemma 4.6, X is hamiltonian. ξ τ ∈ T + τ ∈ T + s.t. τ ∈ T − τ ∈ T − s.t. η , (1 , τ ) ∈ S ∗ η , (1 , τ ) ∈ S ∗ ± , ± , ± , ± ± , ± ± , ± − ± , ± , ± , ± , ± , ± ± , ± , ± , ± − − ± , ± , ± , ± , ± , ± , ± , ± , ± ± , ± , ± , ± −± , ± , ± , ± ± , ± , ± , ± ± , ± ± , ± , ± , ± − ± , ± − ± , ± , ± , ± ± , ± ± , ± , ± , ± ± , ± ± , ± , ± , ± − ± , ± , ± , ± ± , ± , ± , ± ± , ± ± , ± ± , ± ± , ± ± , ± , ± , ± ± , ± , ± , ± ± , ± , ± , ± − ± , ± , ± , ± ± , ± ± , ± , ± , ± ± , ± ± , ± − ± , ± , ± , ± − ± , ± , ± , ± ± , ± ± , ± , ± , ± − ± , ± , ± , ± ± , ± , ± , ± , ± , ± , ± , ± ± , ± − − ± , ± , ± , ± , ± , ± ± , ± , ± , ± ± , ± − ± , ± , ± , ± ± , ± Table 7:
Information about existence of edges in X (PSL(2 , , D , S + ξ ) where ξ ∈ S ∗ ∩ S ∗ + 1, ξ = , With the approach used in Examples 7.13 and 7.14 we will now prove existence of Hamiltoncycles in any basic orbital graph arising from Rows 8 and 9 of Table 4.
Proposition 7.15
Let X = X ( G, H, W ) , where W ∈ { S + ξ , S − ξ } , ξ ∈ S ∗ ∩ S ∗ + 1 and ξ = , , beone of the graphs arising from Row 8 or 9 of Table 4. Then X is hamiltonian. Proof.
Because of the isomorphism given in Proposition 7.4 we can assume that W = S + ξ . Notethat X is of valency ( p − / d ( V ∞ ) = 0, d ( V ∞ , V x ) = 2 for every x ∈ S ∗ , and d ( V ∞ , V x ) = 0 forevery x ∈ N ∗ . The number of edges d ( V y , V x ), x, y ∈ F ∗ , between V y and V x in X ρ is obtained44rom (20) and (21): j , = 12 ( y − x ± p x + 2(1 − ξ ) xy + y ) , (23) j , = 12 ( y − x ± p x − − ξ ) xy + y ) , (24) η , = y ξx ((2 ξ − x − y ± p x + 2(1 − ξ ) xy + y ) , (25) η , = y − ξ ) x ((1 − ξ ) x − y ± p x − − ξ ) xy + y ) . (26)Note that the values of j , , , and η , , , for edges inside V x are j , ( x, x ) = ± x p − ξ, (27) j , ( x, x ) = ± x p ξ, (28) η , ( x, x ) = xξ ( ξ − ± p − ξ ) , (29) η , ( x, x ) = x − ξ ( − ξ ± p ξ ) . (30)Therefore it depends solely on ξ whether there are edges inside the orbit V x or not. In particular,we have that d ( V x ) ∈ { , , } for x ∈ F ∗ . (31)Suppose that x = 1. Then we get from (23) - (26) the following values for the above quantities j and η : j , = 12 ( y − ± p − ξ ) y + y ) ,j , = 12 ( y − ± p − − ξ ) y + y ) ,η , = y ξ ((2 ξ − − y ± p − ξ ) y + y ) ,η , = y − ξ ) ((1 − ξ ) − y ± p − − ξ ) y + y ) . Let us now consider elements of the form 1 + 2(1 − ξ ) g + g and 1 − − ξ ) g + g , where g is a generator of F ∗ . Since existence of Hamilton cycles in X for p ∈ { , , } is proved inExamples 7.12, 7.13 and 7.14, we may assume that p
6∈ { , , } . Consequently, Theorem 3.1and Proposition 3.7 combined together imply that for the two polynomials f ( z ) = 1 + 2(1 − ξ ) z + z and h ( z ) = 1 − − ξ ) z + z there exist g, g ′ ∈ F ∗ such that F ∗ = h g i = h g ′ i and f ( g ) , h ( g ′ ) ∈ S ∗ ∪ { } . It follows that V isadjacent to V g and to V ( g ′ ) in X P depending on whether the corresponding values η , and η , are squares or not. Let s = g and s ′ = ( g ′ ) . Then s and s ′ both generate S ∗ . We claim thateither V is adjacent to V s or V g is adjacent to V sg , and moreover either V is adjacent to V s ′ or V g ′ is adjacent to V s ′ g ′ . For this purpose we need to calculated the corresponding values of η , η ,45 , and η for the following pairs of vertices ( V , V s ), ( V g , V gs ), ( V , V s ′ ), ( V g ′ , V g ′ s ′ ): η , (1 , ¯ s ) = ¯ s ξ ((2 ξ − − ¯ s ± p − ξ )¯ s + ¯ s ) ,η , (1 , ¯ s ) = ¯ s − ξ ) ((1 − ξ ) − ¯ s ± p − − ξ )¯ s + ¯ s ) ,η , (¯ g, ¯ g ¯ s ) = ¯ gη , (1 , ¯ s ) ,η , (¯ g, ¯ g ¯ s ) = ¯ gη , (1 , ¯ s ) , where ¯ s ∈ { s, s ′ } and ¯ g ∈ { g, g ′ } . Hence, η i (¯ g, ¯ g ¯ s ) η i (1 , ¯ s ) = ¯ g ∈ N ∗ , i ∈ { , , , } . Consequently, for each i exactly one of η i (¯ g, ¯ g ¯ s ) and η i (1 , ¯ s ) belongs to S ∗ , implying that in thebicirculant X ρ − { V ∞ } we have a full induced cycle preserved by the automorphism σ either inthe orbit O ( S ∗ ) or in the orbit O ( N ∗ ). We claim that X ρ − { V ∞ } contains a generalized Petersengraph. In order to prove this claim we only need to show that if there is an orbit that does notcontain a full cycle preserved by σ then it contains a union of cycles (preserved by σ ), and thatthe bipartite graph between the two orbits contains a matching preserved by σ . The latter holdssince X is connected and since V ∞ is adjacent only to vertices in O ( S ∗ ) (see Table 4). Supposetherefore that one of the two orbits O ( S ∗ ) and O ( N ∗ ) does not contain a full cycle.Suppose first that O ( N ∗ ) does not contain the above mentioned full cycle. Then V is adjacentto both V s and V s ′ , implying that val ( X ) = p − ≥ d ( V ∞ , V ) + 4 + d ( V ) + | X d ( V , V x ) : x ∈ N ∗ | . = 6 + d ( V ) + | X d ( V , V x ) : x ∈ N ∗ | . On the other hand, since, by assumption the valency of the graph induced on O ( N ∗ ) is either 0or 1, we have by calculating valency val ( X ) at V x , x ∈ N ∗ : val ( X ) = p −
12 = ǫ + d ( V x ) + | X d ( V x , V y ) : y ∈ S ∗ | . where ǫ ∈ { , } . Since | P d ( V , V x ) : x ∈ N ∗ | = | P d ( V x , V y ) : y ∈ S ∗ | it follows that d ( V x ) > O ( S ∗ ) does not contain the above mentioned full cycle. Similarly as aboveit follows that each V x , x ∈ N ∗ , has at least 4 neighbors in X hO ( N ∗ ) i . This implies that val ( X ) = p − ≥ d ( V x ) + | X d ( V x , V y ) : y ∈ S ∗ | . On the other hand, since, by assumption the valency of the graph induced on O ( S ∗ ) is either 0or 1, we have by calculating valency val ( X ) at V x , x ∈ S ∗ : val ( X ) = p −
12 = d ( V ∞ , V ) + ǫ + d ( V ) + | X d ( V , V y ) : y ∈ N ∗ | = 2 + ǫ + d ( V ) + | X d ( V , V y ) : y ∈ N ∗ | , ǫ ∈ { , } . It follows that 2 + ǫ + d ( V ) ≥ d ( V x ), and so d ( V ) ≥ − ǫ + d ( V x ). Since,by (31), d ( V ) and d ( V x ) are both even numbers smaller than or equal to 4, it follows that either d ( V ) = 4 and d ( V x ) = 2 or d ( V ) = 2 and d ( V x ) = 0. None of these is possible. In particular, if d ( V ) = 4 then, by (27) - (30), we have1 ξ ( ξ − ± p − ξ ) ∈ S ∗ and 11 − ξ ( − ξ ± p − ξ ) ∈ S ∗ , and thus η , ( x, x ) = xξ ( ξ − ± p − ξ ) ∈ N ∗ and η , ( x, x ) = 11 − ξ ( − ξ ± p − ξ ) ∈ N ∗ , implying that d ( V x ) = 0. Similarly, if d ( V ) = 2 then two of the above expressions for η i (1 , i ∈ { , , , } , are squares and the other two are non-squares, which implies that either η , ( x, x )or η , ( x, x ) is a square, and consequently d ( V x ) = 2.These contradictions show that the valencies of both X hO ( S ∗ ) i and X hO ( N ∗ ) i are at least 2,which proves that X P − { V ∞ } contains a generalized Petersen graph GP (( p − / , k ) as claimed.If GP (( p − / , k ) is not isomorphic to GP ( n,
2) with n = ( p − / ≡ X P − { V ∞ } . Of course, this cycle containsan edge of the form V x V y , x, y ∈ S ∗ . Replacing this edge with a path V x V ∞ V y gives a Hamiltoncycle in X ρ . Obviously this Hamilton cycle contains double edges, and so, by Lemma 4.6, it liftsto a Hamilton cycle in X . We may therefore assume that ( p − / ≡ X P − { V ∞ } is isomorphic to GP (( p − / , X P − { V ∞ } with both endvertices in O ( S ∗ ).By joining these two endvertices with V ∞ we can then extend this path to a Hamilton cycle in X ρ which lifts to a Hamilton cycle in X . This completes the proof of Proposition 7.15. Proof of Theorem 1.2.
Let X be a connected vertex-transitive graph of order pq , where p and q are primes and p ≥ q , other than the Petersen graph. If q ∈ { , p } then X admits a Hamiltoncycle by Proposition 4.1. We may therefore assume that q
6∈ { , p } . Then X is a generalizedorbital graph arising from one of the actions given in Theorem 4.2. If X is imprimitive then itadmits a Hamilton cycle by Proposition 4.3. We may therefore assume that X is primitive, andso X is a generalized orbital graph arising from one of the group actions given in Table 3. In fact,as explained in Section 4, we can assume that X is a basic orbital graph arising from a groupaction given in Table 3.If X arises from one of Rows 1, 2 and 3 of Table 3 then it admits a Hamilton cycle by Propo-sition 5.1. If X arises from the group action given in Row 4 of Table 3 then it admits a Hamiltoncycle by Proposition 5.2. If X arises from the group action given in Row 5 of Table 3 then itadmits a Hamilton cycle by Propositions 6.7 and 6.8. If X arises from the group action given inRow 6 of Table 3 then the existence of a Hamilton cycle follows from Propositions 7.8, 7.9, 7.10,7.11 and 7.15. Finally, if X arises from the group action given in Row 7 of Table 3 then theexistence of a Hamilton cycle follows from Proposition 5.3.47 cknowledgements The authors wish to thank Marston Conder and Ademir Hujdurovi´c for helpful conversationsabout the material of this paper.
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