aa r X i v : . [ m a t h . C O ] N ov Page 1 of 22 Jacob Steinhardt
Cayley graphs formed by conjugate generating sets of S n Jacob SteinhardtThomas Jefferson High School for Science and Technology [email protected] age 2 of 22 Jacob Steinhardt
Abstract
We investigate subsets of the symmetric group with structure similar to that of a graph. The“trees” of these subsets correspond to minimal conjugate generating sets of the symmetric group.There are two main theorems in this paper. The first is a characterization of minimal conjugategenerating sets of S n . The second is a generalization of a result due to Feng characterizingthe automorphism groups of the Cayley graphs formed by certain generating sets composed ofcycles. We compute the full automorphism groups subject to a weak condition and conjecturethat the characterization still holds without the condition. We also present some computationalresults in relation to Hamiltonicity of Cayley graphs, including a generalization of the work onquasi-hamiltonicity by Gutin and Yeo to undirected graphs. Key words:
Cayley graph, automorphism, transposition, cycle, conjugate, Hamiltonian cycle.age 3 of 22 Jacob Steinhardt
In this paper, we will let N denote the set { , , . . . , n } . S n will denote the symmetric group actingon n elements with canonical action on N . A n will denote the alternating group acting on N . Wewill use the notation ( a a . . . a k )( b b . . . b k ) . . . to express a permutation as a product ofdisjoint cycles. By the support of a permutation we will mean those elements not fixed by thepermutation. Given permutations σ , τ , στ denotes τ ◦ σ .Given a multiset A = ( a , . . . , a k ) with a i ≥
2, we define its extended conjugacy class in S n tobe the set of all permutations such that, when decomposed into disjoint cycles, contain cycles oflengths a , . . . , a k , and no others. We denote it by C ( A ).Given a set S ⊂ N , define the subsymmetric group of S as the set of all permutations in S n thatfix all elements outside of S . Define the subalternating group of S as the set of all even permutationsin S n (i.e., the permutations in A n ) that fix all elements outside of S . A semisymmetric group of S is defined as a subgroup of S n that stabilizes S whose restriction to S forms a symmetric groupacting on S . A semialternating group of S is defined similarly.Given a graph Γ, we let V (Γ) and E (Γ) denote the vertices and edges of Γ, respectively. An Eulerian path is a walk in Γ that traverses each edge exactly once. It is called an
Eulerian cycle if the first and last vertices in the walk are the same. Given a group G and a set S ⊂ G , the Cayley graph
Γ =
Cay ( G, S ) is defined as follows: each vertex is an element of G , and two vertices g, h ∈ V (Γ) are adjacent if gh − ∈ S or hg − ∈ S . Cayley graphs are of general interest in the field of Algebraic Graph Theory and also have certainproperties desirable in practical applications. We present here a brief survey of some of the broaderresults and conjectures surrounding Cayley graphs. Godsil and Royle [8] provide a useful overviewof work on graphs with transitive permutation groups in general, which we partially reproduce here.First, all Cayley graphs are vertex-transitive since the mapping φ g ( x ) = xg is an automorphism forall g ∈ G . As such, there is always a representation of G in Aut ( Cay ( G, S )), denoted R ( G ). R ( G )acts not only transitively but regularly on the vertices of Cay ( G, S ). Sabidussi has shown thatthe converse of this is true, namely that Γ if a Cayley graph of G if and only if Aut (Γ) contains aage 4 of 22 Jacob Steinhardtsubgroup isomorphic to G that acts regularly on V (Γ) [19].Minimally generated Cayley graphs have provably maximal vertex connectivity [8], which pointsto uses in practical applications. Specifically, Cayley graphs have been used to create networks withsmall diameter and valency and high connectivity for uses in parallel processing, and Schreier cosetgraphs, a generalization of Cayley graphs, have been used to solve certain routing problems [3].See also [2] and [20].One tempting conjecture related to Cayley graphs is that every Cayley graph has a Hamiltoniancycle, first given by Strasser in 1959 [18]. For more information, see [4], [12], and [16]. While weoffer some computational ideas in relation to this conjecture, the focus of this paper is on anotherdifficult problem in Algebraic Graph Theory, that of characterizing the automorphism groups ofCayley graphs. We have a poor understanding of the automorphism groups of Cayley graphs,though there are some notable exceptions (see below). These groups are fundamental as the mostnatural algebraic structure to associate with an arbitrary highly symmetric graph.Note that a graph can be defined as a collection of vertices and edges. Two vertices are adjacentif there exists an edge connecting them, and two vertices v and v are connected if there existsa sequence of adjacent vertices containing v and v . On the other hand, consider the followingdefinition: Given a collection of vertices V and a collection of edges E , we can let each elementof E act on V as a transposition swapping the two vertices on which E is incident. If we then letmultiplication in E extend through the definitions of a group action, E generates a subgroup ofthe symmetric group acting on V (we denote this subgroup as < E > ). Then we say that v and v are adjacent if ( v v ) ∈ E , and that v and v are connected if ( v v ) ∈ < E > . Additionally,connected components correspond to orbits of V under E . A tree is a minimal generating set of S | V | consisting only of transpositions (thus the fact that trees have n − n − S n ). It is easily verified that these definitionsare equivalent.The algebraic properties of the related Cayley graphs of trees in the above definition are well-understood. We know in particular that these graphs are Hamiltonian [11]. Furthermore, in 2003Feng [5] generalized a result by Godsil [8] that fully characterizes the automorphism groups of thesegraphs.age 5 of 22 Jacob Steinhardt C -trees The above definition of a graph in terms of transpositions can be generalized. Given a collectionof vertices (which, from now on, for convenience, will without loss of generality be N ), and a set T ⊂ S n in which all elements of T are conjugate (say with conjugacy class C ), then we can defineelementary notions in a C -graph as follows. v , v ∈ N are adjacent if they have the same orbitunder a single element of T . They are semi-connected if they have the same orbit under T , and connected if ( v v ) ∈ < T > (it is then easy to verify that semi-connectivity and connectivity areequivalence relations). Connected components correspond to subsymmetric groups of < T > . A tree is a minimal generating set of S n with all elements lying in C . It is natural to ask why weadd the somewhat artificial-looking stipulation that all elements of T belong to the same conjugacyclass. The main reason is that this stipulation is inherent in the construction of a normal graph,where all edges are transpositions. Additionally, without this restriction we get the result that atree, under our fairly intuitive definition, almost always has 2 edges since (1 2) and (2 3 4 . . . n )generate S n .In this paper we will characterize C -trees and study some of their properties, including a gener-alization of Feng’s result. However, we will still use the language of graphs for the sake of intuition.For approaches to extending the above intuitive generalization to a well-structured system, see theconcluding section on open problems.With C -trees defined, we introduce some more terminology associated to them. A set T ⊂ S n issaid to be semi-connected if N has a single orbit under T (i.e. all elements of N are semi-connected).We call it split if the intersection of the supports of any two elements of T has size at most one.Note that if T generates S n then it must be connected.Given a multiset A and integer n , we define f ( A, n ) to be the infimum of | G | across all G ⊂ C ( A )that generate S n . We aim to find f ( A, n ) for every A for sufficiently large n . Let c ( A ) be definedas | A | X i =1 a i − X ( A ) such that, for n ≥ X ( A ), f ( A, n ) is equalto ⌈ n − c ( A ) ⌉ (1)when c ( A ) is odd, and ∞ otherwise. When c ( A ) is odd, then A defines the conjugacy class ofan even permutation and so f ( A, n ) is obviously ∞ (because it is impossible to generate any oddpermutations). Note further that f ( A, n ) is necessarily at least the value given by (1), as c ( A )counts the number of transpositions necessary to generate an element of C ( A ), and so if it wassmaller then it would be possible to generate S n with less than n − S n . We study first the case of a single k -cycle, i.e. | A | = 1 and a = k . We will give explicit generatorsfor S k − : Proposition 3.1
The set { (1 2 . . . k ) , ( k k + 1 . . . k − } generates S k − when k is even. Proof
We construct something similar to a semisymmetric group of { , , . . . , k } , except with theelements lying in the positions { k, k + 1 , . . . , k − } . From this we will construct a subsymmetricgroup of { , , . . . , k } , which will finally allow us to construct the entire symmetric group S k − . Lemma 3.2
We can place the elements { , , . . . , k } in any order in positions { k, k + 1 , . . . , k − } as long as we allow the other elements to move arbitrarily (even for odd k ). Proof
Let σ = (1 , , . . . , k ) and τ = ( k, k + 1 , . . . , k − k − k elements to any permutation π = ( π , π , . . . , π k ) of (1 , , . . . , k )as follows: Rotate π k to the k th position using σ , then apply τ once. Now rotate π k − to the k thposition (again with σ ), and apply τ again. Continue this process until we have put all k of thedesired elements into place. For example, to make the last 4 elements (2 , , , } , we would apply σ τ σ τ σ τ σ . Each successive set of applications of σ moves the next desired element into place(3, then 1, then 4, then 2).age 7 of 22 Jacob Steinhardt Lemma 3.3
For k even, the k -cycles generate S n for n ≥ k . Moreover, for k odd, the k -cyclesgenerate A n . Proof
Let U be the set of all k -cycles. U is closed under conjugation, so < U > is normal in S n . Thus < U > is either ( e ), D , A n , or S n , where by D we mean the normal subgroup of A isomorphic to the dihedral group of order 4. It can’t be ( e ) because U is non-trivial, and it can’tbe D because D contains no cycles. Thus it is A n or S n . If k is odd, it must be A n because U consists of only even permutations. If k is even, it must be S n since U contains an odd permutation. Lemma 3.4
We can generate the subsymmetric group on { , , . . . , k } when k is even. Proof
Take some such permutation π generated in the manner of Lemma 3.2, and consider πτ π − .This creates an arbitrary k -cycle among the first k elements while fixing the last k − { , , . . . , k } .Now, to generate an arbitrary permutation π = { π , . . . , π k − } in S k − , first use τ to move { π k +1 , . . . , π k − } to the first k − k th position, then, since we can generate any permutation among the first k elements (recallthat we can do this by Lemma 3.4), move it to an arbitrary place among the first k − { π k +1 , . . . , π k − } in the proper order (though leaving them in the positions { , , . . . , k − } ). Wecan then move them to their proper locations with ( στ ) k − . Now that the last k − k elements in place. Wecan thus generate an arbitrary permutation and therefore S k − . This completes Proposition 3.1. Proposition 3.5 f (( k ) , n ( k −
1) + 1) = n for n ≥ and k even. Proof
This follows by induction on n . Proposition 3.1 proves the base case of n = 2. The inductivestep is completed by the following easily verified lemma: Lemma 3.6
The subsymmetric group on S , together with the cycle ( a , . . . , a n ) , generates thesubsymmetric group on S ∪ { a , . . . , a n } provided that S ∩ { a , . . . , a n } 6 = ∅ and S
6⊂ { a , . . . , a n } . age 8 of 22 Jacob Steinhardt Corollary 3.7 f (( k ) , n ) = ⌈ n − k − ⌉ for n ≥ k − . Proof
Take the construction for when n − k − is an integer (i.e. that given above in Proposition 3.5).Then, to extend the formula to arbitrary n , add the k -cycle ( n − k + 1 , n − k + 2 , . . . , n ) and applyLemma 3.6. This completes the claimed characterization of f ( A, n ) for cycles.
Having proven our result for cycles, we would like to extend it to more complex permutations. Wewill start with the simplest of these, i.e. products of disjoint transpositions. We call a permutation basic if it is of this form, and denote B k = (2 , , . . . ,
2) ( k twos). Proposition 3.8 f ( B k , n ) = ⌈ n − k ⌉ for n ≥ k (2 k + 1) + 1 and k odd. Proof
Our goal will be to write 2 k + 1 explicit generators for S k (2 k +1)+1 . Then we can easilyproceed by induction as before. Such generators must form a semi-connected set. However, wewould also like the set to be split so that only one cycle interacts at a time when multiplyingpermutations. To do this, we will find an Eulerian cycle of K k +1 , which will allow us to create aconnected and split set.First note that every vertex of K n +1 has even degree (in particular, degree 2 n ), so that K n +1 has an Eulerian cycle. We now construct generators from this cycle. We start with an example,then give a general method. Consider k = 3, 2 k + 1 = 7, and the cycle 1 → → → → → → → → → → → → → → → → → → → → →
1. We have the generators g = (1 2)(5 6)(12 13) g = (2 3)(9 10)(19 20) g = (6 7)(16 17)(20 21) g = (3 4)(13 14)(17 18) g = (10 11)(14 15)(21 22) g = (7 8)(11 12)(18 19) g = (4 5)(8 9)(15 16)Note that if we follow the path of permutations containing (1 , , (2 , , (3 , , . . . , (21 , g , g , g , g , . . . , g , g , i.e. exactly the constructed Eulerian cycle (with the exception of thefinal vertex). This is the general method in which we will construct our generators. Specifically, weplace the transposition ( i i + 1) in the generator corresponding to the i th vertex visited in the cycle.age 9 of 22 Jacob SteinhardtNote that the properties of an Eulerian cycle guarantee that these generators will be semi-connectedand split. The semi-connected part is obvious, whereas the split part is a consequence of the factthat every edge is traversed exactly once, which corresponds to the fact that each pair of generatorsmove at most one common element. We now show this construction generates S k (2 k +1)+1 . Theorem 3.9 If T ⊂ C ( B k ) is semi-connected and split, then < T > = S n or A n , depending onwhether k is even or odd. Proof
We call two generators adjacent if they move a common element. Consider two adjacentgenerators, g i and g j , and consider g i g j g i g j . All transpositions are applied twice in this case andtherefore cancel, except for the two transpositions that act on the same element, which multiply toa three cycle. So, in the above example, g g g g = (14 15)(15 16)(14 15)(15 16) = (14 16 15) =(14 15 16). In this manner, we generate all 3-cycles of the form ( i i + 1 i + 2). We wish to showthat these generate A n . From this, we would be done, since any odd permutation then allows usto generate S n . In fact, it is convenient for later purposes to prove a slightly stronger result: Lemma 3.10
When n is odd, the subalternating group on S , together with the cycle ( a , . . . , a n ) ,generates the subalternating group on S ∪ { a , . . . , a n } provided that S ∩ { a , . . . , a n } 6 = ∅ and that | S ∩ { a , . . . , a n }| ≤ | S | − . When n is even, it generates the entire subsymmetric group. Proof
Like Lemma 3.6, the proof is easy enough to omit. The only important detail to note isthat we get | S ∩{ a ,...,a n }| !2 distinct permutations, which must be the alternating group when n is oddor must generate the symmetric group by Lagrange’s theorem when n is even.In particular, a 3-cycle looks like A , so the given 3-cycles indeed generate the alternating group(they are semi-connected since T was semi-connected), and we are done with Theorem 3.9.Our proof of the remainder of Proposition 3.8 (i.e. the induction and extension to cases when c ( A ) does not divide the n −
1) follows in exactly the same manner as that of Proposition 3.1, andso we omit it, instead referring the reader to Proposition 3.5 and Corollary 3.7. The only necessarymodification is that we must deal with each of the cycles in the added permutation one at a timein our inductive step.age 10 of 22 Jacob Steinhardt
We would next like a general criterion for connectedness. We present it here:
Definition
A set T ⊂ C ( A ) is called balanced if it is possible to divide the set of orbits of elementsof T into disjoint sets S , S , . . . , S | A | such that all orbits in S i have the same size and each elementof S i overlaps with at least one other element of S i . We will denote the size of the orbits in S i by a i . Theorem 3.11
All semi-connected, split, balanced sets in the extended conjugacy class of an oddpermutation generate S n . Proof
We proceed by induction on two quantities: first | A | , then the number of occurrences of2 in A . Note that we have already proven the base cases of this induction in Corollary 3.7 andProposition 3.8.We call two permutations i -adjacent if they both have orbits in S i . Pick i such that a i ismaximal, and consider any two i -adjacent permutations, σ and τ , with orbits i σ and i τ in S i . By thesame argument as Proposition 3.1, these generate the semialternating group on the elements movedby the two identified cycles (moving 2 a i − Chebyshev’sTheorem [17], there exists a prime strictly between a i and 2 a i , and the semialternating groupcontains a cycle of this length, call it p .Consider each cycle of this length in our semialternating group, and apply it lcm a ∈ A a times.Since p is prime and a j < p for each j , we end up with a p -cycle, which we can then apply somenumber of times to get back to our original p -cycle. Note, however, that all other elements thatwere moved contained only cycles of length a j for some j , and so were all cancelled out by the aboverepeated application. Thus we are left only with the actual p -cycle. Repeating this for all such p -cycles in the semialternating group gives us all actual p -cycles, i.e. those living in the associatedsubalternating group. Thus, by the same arguments as in Lemma 3.3, they generate the entiresubalternating group. If a i is odd, then i σ and i τ both live in this subalternating group, and so wecan take σ ( i σ ) − and τ ( i τ ) − . Taking σ ( i σ ) − for all σ ∈ T (we can do this since T is balanced)gives us a semi-connected, split, balanced set with strictly less orbits in each permutation, so thatwe can apply the inductive step to generate the subsymmetric group on the elements moved byage 11 of 22 Jacob Steinhardtthese new permutations. Then, by adding i σ for each σ ∈ T , by Lemma 3.5 we can generate theentire symmetric group, and we are done.On the other hand, if a i is even, then we can only cancel i σ down to a transposition. However,this gives us an extended conjugacy class with the same number of orbits, but with strictly moreoccurrences of 2 in A than before. Thus we can apply the inductive step in the same manneras above, and are once again done. Note that in both cases we attained elements in the newextended conjugacy class by multiplying elements in the old extended conjugacy class by an evenpermutation. This shows that the new extended conjugacy class does indeed correspond to an oddpermutation.We are now ready to prove our major contention: Theorem 3.12
Let A be a multiset of size k . Then there exists some X ( A ) such that f ( A, n ) = ⌈ n − c ( A ) ⌉ when n ≥ X . Furthermore, X (( k )) ≤ k − , X ( B k ) ≤ (cid:0) k +12 (cid:1) + 1 , and X ( A ) ≤ c ( A )Φ | A | (2 | A | ) + 1 , where Φ k denotes the k th cyclotomic polynomial. Proof
Note that the first two bounds have already been proven. For the final case, we again useEulerian cycles, this time with the goal of creating a semi-connected, split, and balanced set. Inparticular, we find a prime congruent to 1 mod 2 k . We know that such a prime exists that is lessthan Φ k (2 k ) (proof sketch in appendix).Take such a prime, p , fitting the properties described above. If p − = kn , then we will work inthe extended conjugacy class that is equivalent to n copies of A , then use this to move down to A itself. We construct an Eulerian cycle for K p as follows. The edges (mod p ) will be1 , , . . . , p, , , . . . , p, , . . . , p, . . . , p − , p − , . . . , p ( p − p = 7 then we have (in the case of 2-cycles) the associated generators(1 2)(11 12)(19 20) (2 3)(8 9)(17 18)(3 4)(12 13)(15 16) (4 5)(9 10)(20 21)(5 6)(13 14)(18 19) (6 7)(10 11)(16 17) (7 8)(14 15)(21 22)age 12 of 22 Jacob SteinhardtWe can extend this past 2-cycles (for example, permutations in the extended conjugacy class (2 , , i th column until the cycles in that column havelength a i . Note also that this is a balanced set by construction. It is easy to verify that this alsodefines an Eulerian cycle, and is thus connected and split. On the other hand, we have the followingresult: Lemma 3.13 If B is equivalent to k copies of A , and if there exists T ⊂ C ( B ) that generates S n ,then there exists T ′ ⊂ C ( A ) that generates S n , and furthermore such that | T ′ | = k | T | . Proof
Split each σ ∈ T into k permutations such that each of these permutations lies in C ( A ).These obviously generate S n since products of them generate S n .This proves the base case of a final induction showing that f ( A, n ) = ⌈ n − c ( A ) ⌉ for all n ≥ X ,where X = pc ( A ) + 1. This induction will finally prove Theorem 3.12. However, once again thisnew induction is identical to the completions of Propositions 3.1 and 3.8, and so we refer thereaders there for the completion of the proof. We devote this section to the characterization of the automorphism groups of certain C -graphs. Definition
Given a split set of cycles T ⊂ S n , the cycle graph Cyc ( T ) is formed by associatingeach vertex with an element of N and drawing edges x x , x x , . . . , x k − x k if ( x x . . . x k ) is in T . Note that this involves arbitrarily choosing a “starting” and “ending” point for each cycle in T .When T consists of transpositions, Feng [5] refers to Cyc ( T ) as T ra ( T ). Definition
Given a split set of cycles T , the degree of some t ∈ T is defined as the number ofdistinct points in its support that overlap with other cycles. If t has degree 1, we call it a leaf .age 13 of 22 Jacob Steinhardt Definition
A split set of more than two cycles generating S n is said to be normal if any element isadjacent to at most 1 leaf, and furthermore Cyc ( T ) is a tree (note that this is stronger than askingthat T be a minimal generating set of S n , as it effectively adds the criterion that n ≡ k ),where T consists of k -cycles).We use this to offer a partial generalization to a theorem by Feng [5] that states that Aut ( Cay ( S n , T )) ∼ = R ( S n ) ⋊ Aut ( S n , T ), where Aut ( S n , T ) = { φ ∈ Aut ( S n ) | φ ( T ) = T } , and furthermore that Aut ( S n , T ) ∼ = Aut ( T ra ( T )). In the following, T will always be normal, and if we talk about agraph it will be Cay ( S n , T ) unless otherwise specified: Theorem 4.1
Let T be a normal set. Then Aut ( Cay ( S n , T )) ∼ = R ( S n ) ⋊ Aut ( S n , T ) , where R ( S n ) is the representation of S n as an action on Cay ( S n , T ) . Proof
We use Feng’s idea of finding cycles that force graph automorphisms to be group automor-phisms. Certain lemmas requiring case analysis will be dealt with in the appendix.
Lemma 4.2
Let t , t ∈ T . Then there exists a unique -cycle containing the path t → ( e ) → t iff t t = t t , and furthermore the cycle will be ( e ) → t → t t → t → ( e ) . Proof
See appendix.
Lemma 4.3
Let t , t ∈ T such that t t = t t . Then the -cycle corresponding to t t t t t t issent to another cycle of this form under graph automorphisms when t and t are transpositions.Otherwise, the same statement holds for the -cycle corresponding to t t t − t − t t t − t − t t t − t − . Proof
The case of transpositions was dealt with by Feng [5]. It is easily verified that the latterconstruction is a cycle when t and t are not transpositions (it is the union of two cycles when theyare transpositions). Also note that no two consecutive edges correspond to commuting generators,and this property is preserved through graph automorphisms by Lemma 4.2. It is natural to try toprove that this is the only 12-cycle going through t and t where no two consecutive edges commute.However, this is false, as shown by the following counterexample: Let a = (1 2 3 4), b = (1 5 6 7), c = (1 8 9 10), d = (1 11 12 13). Then aba − b − aba − b − aba − b − = abcdcb − a − bc − d − c − b − =( e ). We say that edge types are preserved by an automorphism if, whenver x y and x y are edgesage 14 of 22 Jacob Steinhardtcorresponding to the same element of the generating set, then so are φ ( x y ) and φ ( x y ). If weonly allow use of the symbols a, b, a − , b − and assume that edge types are preserved, then this isindeed the only noncommuting 12-cycle, as demonstrated in the appendix. This leads to a proofof our theorem in a special case, which we will make use of: Lemma 4.4
Theorem 4.1 holds when | T | = 2 , assuming that edge types are preserved. Proof
The preceding comments show us that commutators of generators map to commutatorsof generators. Thus φ ( a ) φ ( b ) = φ ( ab ) for all generators a, b , so that φ ( x ) φ ( y ) = φ ( xy ) for all xy by induction. The induction itself is sufficiently non-trivial that we feel obliged to includeit, but sufficiently technical that we will relegate it to the appendix, even though it requires nocase analysis. We have thus shown that all graph automorphisms fixing ( e ) are in fact groupautomorphisms as well. That this implies Theorem 4.1 we wait to prove in full generality at theend of this section.Now for any a, b ∈ T , look at Γ = Cay ( S n , { a, b } ) ⊂ Γ. The 12-cycle described above mustlie inside Γ . We wish to show that, for any automorphism φ ∈ Aut (Γ) fixing ( e ), φ (Γ ) = Cay ( S n , { φ ( a ) , φ ( b ) } ), from which it will follow that commutators map to commutators in general,and we will have proved Lemma 4.3, whence Theorem 4.1 follows from the same arguments as inLemma 4.4.We will, in fact, prove a stronger contention, namely that if two edges represent the same groupelement, then their images also represent the same group element. We first offer an automorphism-invariant criterion for determining whether two adjacent edges represent the same group elementof the Cayley graph when T is normal. Lemma 4.5
Let x → y → z be a path in Γ . Then xy and yz represent the same group elementif and only if the number of -cycles going through xy equals the number of -cycles going through yz . Proof
Note that if xy and yz correspond to the same group element, then the number of 4-cyclesgoing through xy definitely equals the number of 4-cycles going through yz by Lemma 4.2. (Notethat this is true even if T consists of 4-cycles.) The opposite direction is an easy consequence ofthe normality condition.age 15 of 22 Jacob SteinhardtNow note that, by looking at commutativity of edges, we obtain the incidence structure of Cyc ( T ). Thus the group elements that each edge corresponds to is uniquely determined by whichedge each leaf corresponds to (this is simply a consequence of the fact that a tree is determined bythe paths between terminal nodes). Thus, given an edge from v corresponding to a leaf λ , whosepre-image under φ is λ , it suffices to prove that any edge from an adjacent vertex w correspondingto λ also has pre-image λ . First note that unless | T | = 2, which has already been dispatched of,all leaves commute. We consider two cases: adjacency between v and w is induced by a leaf, or theadjacency is induced by a non-leaf. Case one:
We may assume that all leaves commute, whence we are done by Lemma 4.2.
Case two:
By the normality condition, the group element associated with vw must commutewith all but one edge, from which we are again done by Lemma 4.2. Then, noting that leaves aremapped to leaves under any graph automorphism, the final leaf only has one place to go (actually,one could make the argument that there are two places to go – to itself or to its inverse, but bothof these edges correspond to the same group element, which is all that we care about).This completes our contention, so that we are finally done with Lemma 4.3.By Lemma 4.2, commutativity of edges is preserved through graph automorphisms. Further-more, cycles are preserved through graph automorphisms. Thus in particular, { φ ( t ) , φ ( t ) φ ( t ) , φ ( t ) , ( e ) } must be the image of { t , t t , t , ( e ) } if φ is a graph automorphism fixing ( e ) and t , t ∈ T com-mute. This implies that φ ( t ) φ ( t ) = φ ( t t ). By the same argument, and using Lemma 4.3, φ ( t ) φ ( t ) = φ ( t t ) if t , t ∈ T don’t commute. Thus φ ( t ) φ ( t ) = φ ( t t ) for all t , t ∈ T . Thisimplies that φ is not only a graph automorphism but a group automorphism, by the same argumentas in Lemma 4.4.It follows by abuse of notation that Aut ( Cay ( S n , T )) ( e ) ⊂ Aut ( S n , T ), where G x denotes thestabilizer of x under the action of G . But it is well-known that Aut ( S n ) ∼ = S n (the isomorphismbeing with the inner automorphism group) for n = 6 [5], so that Aut ( S n , T ) ⊂ Aut ( Cay ( S n , T )) ( e ) when n = 6 (it is easily verified that any inner automorphism of S n preserving T must alsopreserve incidence in Γ and is thus a graph automorphism). Note that n = 6 only when k = 2,which has already been dispatched, so the theorem holds for all n that we care about. Since Aut ( Cay ( S n , T )) = R ( S n ) Aut ( Cay ( S n , T )) ( e ) and the two subgroups have trivial intersection, weage 16 of 22 Jacob Steinhardtwill have a complete characterization of Aut ( Cay ( S n , T )) if we can show that R ( S n ) is normal.This follows since R ( S n ) is closed under conjugation by elements of Aut ( S n , T ). Thus we have that Aut ( Cay ( S n , T )) ∼ = R ( S n ) ⋊ Aut ( S n , T ), as stated. Comment
Though it is always regrettable when a result cannot be proven in full generality, weclaim that the normality condition is relatively weak. Indeed, given any set T , we can define a normalization of T to be a new C -graph obtained from T by adding another cycle incident on eachleaf of T . It is easily verified that this results in a normal set. Comment
Though non-normal generating sets for S n are too big to test, the characterizationworks for A with the generating set T = { (1 2 3) , (1 3 2) , (1 4 5) , (1 5 4) , (1 6 7) , (1 7 6) } , asthe automorphism group has size 120960 (computed by nauty). In this case the group remains thesemi-direct product of A and the automorphisms of S fixing T . For the sake of future work on the conjecture of Rappaport-Strasser and on Hamiltonicity of directedgraphs in general, we generalize the work of Gutin and Yeo to on quasi-hamiltonicity to undirectedgraphs (see [10] for the original paper). We will assume that R ⊂ V (Γ). Definition A cycle factor in an undirected graph Γ is a subgraph of Γ such that every vertex hasdegree 2. Definition
Let QH (Γ , R ) := { e ∈ E (Γ) | e ∪ R is in a cycle factor } . For k >
1, let QH k (Γ , R ) := { e ∈ E (Γ) | QH k − (Γ , e ∪ R ) is connected } . Then we say that Γ is k -quasi-hamiltonian if QH k (Γ , {} )is connected.Obviously k -quasi-hamiltonicity in an undirected graph implies k -quasi-hamiltonicity in theassociated digraph. Indeed, it is equivalent to k -quasi-hamiltoncity for digraphs if we disallowcycles of length 2 in the cycle factor. In particular, an undirected graph is Hamiltonian iff it is( n − n − Theorem 5.1
Given Γ , define the bipartite graph B (Γ) to have vertex set T = { x , . . . , x m }∪ T = { y , . . . , y m } , where m = | V (Γ) | , and there exists a directed edge from x i to x j iff vertices i and j are adjacent in Γ . Create a flow network where each edge in B (Γ) has capacity and there is asource s with an edge of capacity into every vertex in T , similarly an edge of capacity fromevery vertex in T into a sink t . Then there exists a cycle factor in Γ containing R iff there existsa flow of m from s to t , such that all edges pertaining to elements of R have flow going throughthem. Proof
Suppose that there exists a cycle factor of Γ containing R . Push flow through x i y j and x j y i iff the edge ij is in the cycle factor. This gives the desired flow. Now suppose that we havesuch a flow and wish to construct a cycle factor. It is well-known that we can “force” flow togo through an edge by finding an augmenting path containing that edge and then not adding theback-flow through that edge when we push flow through the augmenting path. Thus asking for theexistence of such a flow is equivalent to forcing flow through all of the edges pertaining to R (forbrevity, from now on we will call this “forcing flow through R ”) and asking for the existence of aflow of 2( m − | R | ) in the resulting graph. Since any choice of augmenting paths must give us thesame maximum flow, we can choose any set of augmenting paths that forces flow through R . Inparticular, given any augmenting path P , we can define another path r ( P ) to be the path obtainedby replacing all instances of x i with y i (and vice versa) and reversing the orientation of each edge in P . Note that P and r ( P ) are edge-disjoint since Γ contains no self-loops. If whenever we augmentby a path P , we also augment by r ( P ), then it will be true that P is an augmenting path iff r ( P )is an augmenting path. In particular, we do this while forcing flow through R . We then continueto do this while performing the maxflow algorithm. By the symmetry of our algorithm, after wehave completed it there will be flow through an edge x i y j iff there is flow through an edge x j y i .Now take the subgraph Γ of Γ formed by all edges ij such that there is flow through x i y j in B (Γ).Since we have a flow of 2 m by assumption, every vertex in Γ has degree 2, thus is a cycle factor,completing the theorem.Due to the high degree of symmetry of Cayley graphs, if Γ does not contain a Hamiltoniancycle then it is (intuitively) likely to have a quasi-hamiltonicity number sufficiently high that it isinfeasible to check. We would thus like a more efficient block to Hamiltonicity for Cayley graphs.age 18 of 22 Jacob Steinhardt Definition
A subset T of a group G is said to have a left coset partition if there exists a set S such that s T and s T are disjoint for distinct s , s ∈ S , and such that ST = G . Definition
A cycle factor is said to be symmetric if it is also a left coset partition.Note that any Hamiltonian cycle is also a symmetric cycle factor. We can thus define theanalogous form of quasi-hamiltonicity where all cycle factors are required to be symmetric. Givena sufficiently crisp characterization of sets with coset partitions, it seems likely that a more effectivealgorithm for Hamiltonicity blocks could be designed.
A minor but interesting detail of this paper is the dependence of our bound on X ( A ) on theexistence of certain primes. There is no reason to believe that this bound should be strict, and so amore complete understanding of C -trees may be reached by a more precise study of the propertiesof X ( A ). If the bound is given by explicit constructions, then the result of such a study wouldalso be smaller Cayley graphs that would be more feasible to analyze empirically.Disregarding our poor understanding of X ( A ), C -trees have been effectively characterized.With this stepping stone, it would be useful to define some more concepts related to C -graphs (ina structurally interesting way). After a tree, the next simplest definition to make is that of a cycle.For a possible idea, we will borrow ideas from matroid theory. We call a set independent if it is asubset of a tree, and dependent otherwise. A simple cycle is then a subset of T that is dependent,but whose proper subsets are all independent. Of course, any other algebraic properties of graphsthat could carry over to C -graphs would also be interesting. The author believes that an alternatedefinition of C -trees leading to a nice matroid structure on the power set of T would make allremaining generalizations transparent. The trees under this structure would also most likely leadto even more structured Cayley graphs.We have fully characterized the automorphism groups of certain C -trees. We would like a gen-eralization, at the very least, of arbitrary split sets consisting of k -cycles. We conjecture thatTheorem 4.1 holds for all such sets. Similarly, we seek a generalization of Feng’s theorem re-garding the isomorphism between Aut ( T ra ( T )) and Aut ( S n , T ) that gives a relation between theage 19 of 22 Jacob Steinhardtautomorphism groups of Cyc ( T ) and Aut ( S n , T ).We turn to the spectral analysis of the Cayley graphs in question. In particular, we look atthe Cayley graphs formed when | T | = 2. Clearly the largest eigenvalue is 4 in this case, sincethe Laplacian is a positive semidefinite matrix. It is interesting to note that the second-largesteigenvalue is 1 + √ k = 2 and 1 + √ k = 4 (the latter result was establishedempyrically). It is therefore very tempting to conjecture that the second-largest eigenvalue isalways 1 + √ k −
1, but this is unfortunately nonsense since it can never exceed 4.Most importantly, this paper points to a deeper connection between Cayley graphs formed bytranspositions and by k -cycles. This is structurally apparent in the similarities between the twoin terms of commutativity and conjugacy, and indicates that more results should generalize to thecase of k -cycles. For example, in [1] the Coxeter representation of the transpositions is used to gaininformation about the spectrum of the Cayley graphs. A generalization of the notion of Coxeterrepresentation to account for k -cycles would likely allow for the generalization of these results.Finally, we propose a more general mathematical program to understand the nature of Cayleygraphs formed by conjugate generating sets in general, which we believe to be a distinguishedvariety. So far, all theorems regarding these graphs show that the automorphism group is minimalin a certain sense. We propose the task of finding cases when the group is not minimal, but is closeto minimal, and analyzing what happens there, when everything should be more transparent. Thisshould point us towards more general results regarding these graphs. Theorem 7.1
For each n > , there is a prime of the form kn + 1 that divides Φ n ( n ) . Proof
It can be shown that if p | Φ n ( j ), then either p | n or p ≡ n ). But Φ n ( n ) ≡ n ),so we must have the second case. Additionally, Φ n ( n ) = Q n − ξ , for each primitive root of unity ξ . But Y n − ξ = qY ( n − ξ )( n − ¯ ξ ) = qY n + 1 − n cos( θ ) > qY n + 1 − n = Y n − ≥ n ( n ), and we are done. -cycles We are asking for a ′ and b ′ such that ab ′ a ′ b = ( e ), or equivalently bab ′ a ′ = ( e ). Thus (for thesupports of ab and a ′ b ′ to be the same) a ′ b ′ ∈ { ab, ab − , a − b, a − b − , ba } . a ′ cannot be b − and b ′ cannot be a − since this would correspond to a path doubling back on itself. Since the productof two split k -cycles is a 2 k − abab is a 2 k − abab − = a ( bab − ) is the product of two k -cycles with different supports, and so is again not theidentity. Similar logic holds for aba − b = ( aba − ) b . aba − b − = ( e ) implies that ab = ba , whichis what we want. Finally, abba = ( e ) implies aabb = ( e ), which is impossible since aa and bb havedifferent supports. The full details of this argument can be found at .The crux of the argument is repeated use of symmetry, which eventually shows that all interestingcases WLOG start with abab − . We then list out all products of four generators such that no twoconsecutive generators commute or represent the same element. By looking at what compositionsof permutations can send 1 back to 1, we reduce essentially to 6 remaining cases, which are easyto check through simple calculations. This completes our case analysis. We have already shown that φ ( a ) φ ( b ) = φ ( ab ) for any automorphism φ of Γ fixing ( e ). We havethe following lemma: Lemma 7.2 If φ is a (graph) automorphism of Γ , then so is φ y = φ ( y − ) φ ( yx ) . The proof is a routine verification. Now, we wish to show by induction that φ ( t t . . . t n ) = φ ( t ) φ ( t ) . . . φ ( t n )for all φ ∈ Γ. Now note thatage 21 of 22 Jacob Steinhardt φ ( t t . . . t n ) = φ ( t ) φ t ( t . . . t n ) = φ ( t ) φ t ( t ) . . . φ t ( t n ) = φ ( t ) . . . φ ( t n )where the equality between the second and third expressions follows by the inductive step. Thiscompletes our induction. The author would like to thank John Dell, Dave Jensen, Alfonso Gracia-Saz, Jim Lawrence, andBrendan McKay for their help, as well as all of the staff of MathCamp 2007 and Thomas JeffersonHigh School for Science and Technology.
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