aa r X i v : . [ m a t h . C O ] S e p Clique-partitioned graphs
Grahame Erskine ∗∗ [email protected] Terry Griggs ∗ [email protected] Jozef ˇSir´aˇn ∗ [email protected] Abstract
A graph G of order nv where n ≥ v ≥ weakly ( n, v ) -clique-partitioned if its vertex set can be decomposed in a unique way into n vertex-disjoint v -cliques. It is strongly ( n, v ) -clique-partitioned if in addition, the only v -cliques of G are the n cliques in thedecomposition. We determine the structure of such graphs which have the largest possiblenumber of edges. In this paper we introduce and solve a problem in extremal graph theory. This concerns whatwe call clique-partitioned graphs, and we consider two versions of the problem. The idea forthis investigation came initially from a BBC TV quiz show “Only Connect”. The premise ofthe quiz is that the teams are required to discover (often obscure) connections between clues.Our starting point is the final round of the quiz, where each team is presented with a 4 × .
14, 22 / /
113 looklike approximations to π . But is this a correct group? If so, should we include 3 in it? Whatabout π itself? A few moments spent looking at the grid will reveal other potential matches.For example (1 + i √ e iπ/ and e i √ are complex numbers. But should we also include − i which is purely imaginary? The numbers 1, 3, 5 and 13 all appear in the Fibonacci sequence.However the crucial feature of the grid is that there is one and only one way to decompose the16 entries into 4 groups of 4, such that each group of 4 has some obvious link. The solution tothis problem is shown at Figure 2.We may model this situation by means of a graph. The vertices of the graph will be the 16clues, and two vertices will have an edge between them whenever there is a plausible link betweenthe corresponding clues. There are two versions of the problem. In the first version, there areexactly four 4-cliques in the graph and the problem is simply to find them. In the second(harder) version, there may be other 4-cliques in the graph (corresponding to sets of four clueswith a plausible link between them). However, there must be only one way to decompose thevertex set of the graph into four of these cliques. ∗ Open University, Milton Keynes, UKMathematics subject classification: 05C35Keywords: extremal graphs i √ − i e i √ π log 2log 3 − + 1 1 227 e iπ/ .
14 2 √ Figure 1: An example quizHowever, our aim in this paper is not just to consider the specific case discussed above, but todeal with the more general problem of graphs which can be decomposed into a set of n cliqueseach of size v , where n ≥ v ≥ Definitions.
Let n ≥ v ≥
2. Let G be a graph of order nv .Then G is weakly ( n, v ) -clique-partitioned if its vertex set can be decomposed in a unique wayinto n vertex-disjoint v -cliques. G is strongly ( n, v ) -clique-partitioned if it is weakly clique-partitioned and in addition, the only v -cliques in G are the n cliques in the decomposition. Question 1.
What is the largest possible number of edges in a strongly ( n, v )-clique-partitionedgraph?
Question 2.
What is the largest possible number of edges in a weakly ( n, v )-clique-partitionedgraph?For both questions we prove an upper bound on the largest possible number of edges in the graph,and construct graphs which attain the upper bounds. We also determine their automorphismgroups. In the case of weakly ( n, v )-clique-partitioned graphs, these graphs are shown to beunique and are a generalisation of a classical result of Hetyei [3] on graphs with a unique perfectmatching. Strongly ( n, v )-clique-partitioned graphs with the largest possible number of edgesare unique for the values n = 2 and v = 2 or 3, and we give enumeration results for some othervalues of ( n, v ). Other links to extremal graph theory are also discussed.First, in Section 2 we deal with strongly clique-partitioned graphs, and then in Section 3 withweakly clique-partitioned graphs. 2 i √ − i e iπ/ Roots of unity .
14 355113 227 − Non-integerrationals π √ e i √ log 2log 3 Transcendentalnumbers
13 3 5 2 + 1 PrimesFigure 2: Solution to example quiz
We begin with a lemma which gives an upper bound on the number of edges in a stronglyclique-partitioned graph.
Lemma 2.1.
Let n ≥ and v ≥ . Let G be a strongly ( n, v ) -clique-partitioned graph. Thenthe number of edges in G is at most n v ( v −
1) + nv ( n − v − . Proof.
The graph G contains n vertex-disjoint v -cliques which contain in total n (cid:18) v (cid:19) edges.Let u be an arbitrary vertex of G . The number of edges from u to any v -clique not containing u is at most v −
2; otherwise we could form a new v -clique, contrary to the strongly clique-partitioned property (See Figure 3). So each of the nv vertices in G can be joined by at most v − n − G is n v ( v −
1) + nv ( n − v − . We shall say that a strongly ( n, v )-clique-partitioned graph attaining this bound is maximal . ItFigure 3: Creating a new clique by adding v − (a) (b) (c)
00 010203 10 11121320 21222330 313233 00 010203 10 11121320 21222330 313233 00 010203 10 11121320 21222330 313233 (d) (e) (f)Figure 4: Construction of the graph Γ(4 ,
4) by adding edgesturns out that maximal graphs do in fact exist. To show this, we define the following graphΓ( n, v ) for any n ≥ v ≥ n, v ) is the set { ( i, j ) : 0 ≤ i ≤ n − , ≤ j ≤ v − } . The adjacency rule is:( i, j ) ∼ ( k, ℓ ) ⇐⇒ i = k and j = ℓi < k and ℓ − j v ) i > k and j − ℓ v )The construction is illustrated in Figure 4 for the case n = 4 , v = 4. In Figure 4(a) the v -cliquesare formed by the first line of the adjacency rules. In Figure 4(b) we add the neighbours ofvertex (0 ,
0) in clique 1 using the second and third lines. In Figure 4(c) we add the neighboursin clique 1 of the remaining vertices in clique 0. In Figure 4(d) we add the neighbours of vertex(0 ,
0) in cliques 2 and 3, and in Figure 4(e) we do the same for the remaining vertices in clique0. Finally in Figure 4(f) we complete the graph by adding the edges between cliques 1 and 2;cliques 1 and 3; and cliques 2 and 3.Our aim is to show that the above graph is strongly ( n, v )-clique-partitioned. We begin with alemma which is a simple consequence of the adjacency relations in Γ( n, v ). Lemma 2.2.
Any v -clique in Γ( n, v ) contains precisely one vertex of the form ( i, j ) for every j ∈ { , , . . . , v − } . Theorem 2.3.
Let n ≥ and v ≥ . Then Γ( n, v ) is strongly ( n, v ) -clique-partitioned andhence the upper bound of Lemma 2.1 is attained.Proof. Let C be a v -clique in Γ( n, v ). Then from Lemma 2.2, the vertices of C are { ( i j , j ) : 0 ≤ j ≤ v − } .We proceed iteratively. Choose x such that i x = min { i j : 0 ≤ j ≤ v − } . Now consider thevertices ( i x , x ) and ( i x +1 , x + 1). By the adjacency relations in Γ( n, v ) it follows that i x = i x +1 .4ext consider the vertices ( i x +1 , x + 1) = ( i x , x + 1) and ( i x +2 , x + 2). Again by the adjacencyrelations in Γ( n, v ) it follows that i x = i x +2 .The argument can now be repeated to get that all i j , j = 0 , , . . . , v − C is one of the v -cliques into which the graph Γ( n, v ) can be strongly clique-partitioned.The algebraic construction of these graphs allows us to deduce some structural information aboutthem. The graphs Γ( n, v ) are clearly regular, of order nv and degree ( v −
1) + ( n − v −
2) = n ( v −
2) + 1, as are any strongly ( n, v )-clique-partitioned graphs meeting the upper bound ofLemma 2.1. In fact it turns out that the graphs Γ( n, v ) are Cayley graphs of a cyclic group, andhence vertex-transitive.Recall that a
Cayley graph
Cay(
G, S ) of a group G and inverse-closed subset S ⊆ ( G \ { } ) hasas vertex set the elements of G , and has edges from g to gs for every g ∈ G and s ∈ S . By awell-known result of Sabidussi [7], a graph is a Cayley graph if and only if it admits a subgroupof automorphisms acting regularly on its vertex set. This regular subgroup is then isomorphicto the group G . A Cayley graph of a cyclic group is often called a circulant graph.For identification purposes, in the following discussion a vertex u = ( a, b ) of Γ( n, v ) will be saidto be in clique number a and have vertex number b . Proposition 2.4.
Let n ≥ and v ≥ . Then Γ( n, v ) is a Cayley graph of the cyclic group oforder nv .Proof. It suffices to exhibit an automorphism of Γ( n, v ) of order nv . Let σ be the cyclic permu-tation of the vertices of Γ( n, v ) defined as follows.(0 , → (1 , → · · · → ( n − , → (0 , v − → (1 , v − → · · · → ( n − , v − → ... → (0 , → (1 , → · · · → ( n − , → (0 , σ has order nv , and maps the vertices of clique i onto those of clique i + 1 (mod n ) forall i = 0 , . . . , n −
1. It remains to show that σ preserves the edges between vertices in differentcliques.Let u = ( a, b ) be a vertex of Γ( n, v ) and let X ( u ) be the set of non-neighbours of u in differentcliques. Let u σ and X ( u ) σ represent the images of u and X ( u ) respectively under σ . To showthat σ is an automorphism we must show that X ( u σ ) = X ( u ) σ for all u .Let U i be the set of non-neighbours of u in clique i = a . If i < a then U i = { ( i, b − , ( i, b ) } andif i > a then U i = { ( i, b ) , ( i, b + 1) } . Under the permutation σ , these sets are mapped to: U σi = { ( i + 1 , b − , ( i + 1 , b ) } if 0 ≤ i ≤ a − { ( i + 1 , b ) , ( i + 1 , b + 1) } if a + 1 ≤ i ≤ n − { (0 , b − , (0 , b ) } if i = n − a = n − X ( u ) σ = [ i = a U σi = { ( k, b − , ( k, b ) : 0 ≤ k ≤ a }∪ { ( k, b ) , ( k, b + 1) : a + 2 ≤ k ≤ n − } If a = n − X ( u ) σ = { ( k, b − , ( k, b ) : 1 ≤ k ≤ n − }
5e now compute X ( u σ ) and show that it is equal to the above expression. Case 1. If a < n − u σ = ( a + 1 , b ). We consider first cliques numbered 0 to a . Theneighbours of u σ in clique i within this range are { ( i, b − , ( i, b ) } . If i is in the range from a + 2to n −
1, then the neighbours of u σ in clique i are { ( i, b ) , ( i, b + 1) } . So: X ( u σ ) = { ( i, b − , ( i, b ) : 0 ≤ i ≤ a }∪ { ( i, b ) , ( i, b + 1) : a + 2 ≤ i ≤ n − } = X ( u ) σ Case 2. If a = n − u σ = (0 , b − u σ in clique i are { ( i, b − , ( i, b ) } for 1 ≤ i ≤ n −
1. So: X ( u σ ) = { ( i, b − , ( i, b ) : 1 ≤ i ≤ n − } = X ( u ) σ So in all cases, X ( u σ ) = X ( u ) σ and so σ is an automorphism of Γ( n, v ) of order nv . Thus h σ i isa group of automorphisms acting regularly on the vertex set of Γ( n, v ), isomorphic to the cyclicgroup of order nv as required.Since Γ( n, v ) is a circulant graph of order nv , its automorphism group must contain a subgroupisomorphic to the dihedral group of order 2 nv . In fact our next result shows that if v ≥
3, thisis the full automorphism group.
Proposition 2.5.
Let n ≥ and v ≥ . Then Aut(Γ( n, v )) is isomorphic to: S ≀ S n ∼ = Z n ⋊ S n , of order n n ! , if v = 2 ;the dihedral group D nv , of order nv , if v ≥ .Proof. If v = 2, then Γ( n, v ) consists of a set of n vertex-disjoint edges, and the result followsimmediately.So suppose v ≥
3. Since Γ( n, v ) is vertex-transitive, we need only show that the stabiliser of anarbitrary vertex has order 2. So consider the stabiliser S of vertex (0 , { (1 , , (1 , } , { (2 , , (2 , } , . . . , { ( n − , , ( n − , } .Consider an automorphism φ ∈ S . We denote the image of a vertex ( i, j ) under φ by ( i, j ) φ .Clearly (0 , φ = (0 , j ) for some j >
0, since any automorphism must preserve the cliques. Inaddition, the non-neighbours { ( i, , ( i, } of (0 ,
0) in clique i (1 ≤ i ≤ n −
1) are mapped setwiseto { ( i ′ , , ( i ′ , } for some i ′ = 0. There are now two cases to consider. Case 1.
If for any i ≥
1, ( i, φ = ( i ′ , i, φ = ( i ′ , ,
1) is alsofixed by φ since it is the only other non-neighbour of ( i ′ ,
1) in clique 0. Similarly, ( i, φ is then( i ′ ,
2) so (0 ,
2) is fixed. Iteratively, we find that clique 0 is fixed pointwise by φ , and each ( i, j )is mapped to ( i ′ , j ), 0 ≤ j ≤ v −
1. Now because ( i,
0) has non-neighbours ( i ′ ,
0) and ( i ′ ,
1) if i < i ′ , and ( i ′ ,
0) and ( i ′ , v −
1) if i > i ′ , the relative ordering of cliques 1 , . . . , v − φ . This can only happen if the numbering of the cliques is fixed by φ , and hence φ is thetrivial automorphism. Case 2.
If for any i ≥
1, ( i, φ = ( i ′ , i . Further, for all i > i, φ = ( i ′ ,
0) and by following the adjacency rules we find that( i, φ = ( i ′ , v − i, φ = ( i ′ , v −
2) and so on. In clique 0, we must have (0 , φ = (0 , v − , φ = (0 , v −
2) and so on. In this case the relative order of the cliques other than 0 is reversedby φ , so that clique 1 is mapped to clique n − φ in this case.Thus S has order 2, and so the full automorphism group Aut(Γ( n, v )) has order 2 nv and henceis isomorphic to the dihedral group D nv . 6e now deal with some specific cases, beginning with n = 2. Theorem 2.6.
The graph
Γ(2 , v ) is the unique maximal strongly (2 , v ) -clique-partitioned graph,and is the complete graph K v with a Hamiltonian cycle removed.Proof. Let G be a strongly (2 , v )-clique-partitioned graph attaining the upper bound. Then byLemma 2.1, G has v ( v −
1) + v ( v −
2) edges and consists of two vertex-disjoint complete graphs K v joined by v ( v −
2) edges. It is therefore the complete graph K v from which 2 v edges havebeen removed. Further, each vertex of the graph G has valency ( v −
1) + ( v −
2) = 2 v − v edges which have been removed are a union of cycles. However, if there is more thana single cycle, then it is clear that the graph would contain a further v -clique, contrary to thedefinition of being strongly clique-partitioned. Hence the edges removed must be a Hamiltoniancycle.The case v = 2 can also be solved completely. Theorem 2.7.
The graph Γ( n, is the unique maximal strongly ( n, -clique-partitioned graph,and consists of n vertex-disjoint edges.Proof. From Lemma 2.1, a graph attaining the bound has n edges. Since the graph must contain2 n vertices, the result follows immediately.Next we deal with the case v = 3. First we need the following definitions and lemma. Definitions. A tournament is a digraph in which every pair of distinct vertices is joined byprecisely one arc. Equivalently, a tournament is a complete graph in which every edge has aspecified orientation. A tournament is acyclic if it contains no directed cycles.It is clear that if a tournament is not acyclic then it must contain a directed 3-cycle. Supposethat a tournament contains a directed m -cycle, ( a , a , . . . , a m ) where m ≥
4. If the tournamentcontains the arc ( a , a ) then it contains a directed 3-cycle. Otherwise it contains the arc ( a , a )and so contains a directed ( m − a , a , . . . , a m ). Repeating this argument, we see thatthe tournament contains a directed 3-cycle.The following lemma is well-known, see for example [1, Theorem 7.13]. Lemma 2.8.
For every n ≥ , there exists an acyclic tournament on n vertices which is uniqueup to isomorphism. We are now in a position to prove the next theorem.
Theorem 2.9.
The graph Γ( n, is the unique maximal strongly ( n, -clique-partitioned graph.Proof. Let G be a strongly ( n, G to be the set { ( i, j ) : 1 ≤ i ≤ n − , ≤ j ≤ } where the cliques are defined bythe adjacencies ( i, j ) ∼ ( i, k ) for all i = 0 , , . . . , n − j, k = 0 , , , j = k . From Lemma 2.1,each vertex is also joined by one edge to each of the n − , j ) ∼ ( i, j + 2) for all i = 1 , , . . . , n − j = 0 , , n = 2, there are no further adjacencies and the graph is Γ(2 , n = 3, there are further adjacencies between clique 1 and clique 2. If (1 , j ) ∼ (2 , j ), j = 0 , ,
2, further cliques would be introduced, so there are two possibilities:7
Figure 5: The graph Γ(2 ,
21 22 2002 00 01 11 12 10 21 22 2002 00 01 11 12 10 (a) (b)Figure 6: Constructing strongly (3 , , j ) ∼ (2 , j + 2) , j = 0 , , , j ) ∼ (2 , j + 1) , j = 0 , , (cid:0) (1 ,
0) (2 , (cid:1)(cid:0) (1 ,
1) (2 , (cid:1)(cid:0) (1 ,
2) (2 , (cid:1) . (See Figure 6.)Therefore, again without loss of generality, we may assume possibility (a). The graph obtainedis Γ(3 ,
3) and is the graph Q24 illustrated in [6, p.145].When n = 4, there are further adjacencies between clique i and clique k , 1 ≤ i < k ≤
3, i.e.( i, k ) ∈ { (1 , , (1 , , (2 , } . Again there are two possibilities:(a) ( i, j ) ∼ ( k, j + 2) , j = 0 , , i, j ) ∼ ( k, j + 1) , j = 0 , , k, j ) ∼ ( i, j + 2) , j = 0 , ,
2. We denote possibility (a)by i → k and possibility (b) by k → i . So up to isomorphism, there are two cases to consider:(i) 1 → , → , → → , → , → ,
3) but case (ii) introduces further cliques, for example the vertices(1 , ,
2) and (3 , n ≥
5. There are further adjacencies between clique i and clique k , 1 ≤ i < k ≤ n − n = 4, these can be determined by a tournamenton the vertex set { i : 1 ≤ i ≤ n − } . However, in order not to introduce further cliques, thetournament must not contain a directed 3-cycle, i.e. it must be acyclic.8hus from Lemma 2.2, a strongly ( n, n ≥
5) attaining the upperbound of Lemma 2.1 is unique and is the graph Γ( n, n = 3 and v = 4. We have the following result. Theorem 2.10.
There are precisely two maximal strongly (3 , -clique-partitioned graphs.Proof. Let the vertex set of a maximal strongly (3 , { ( i, j ) :0 ≤ i ≤ , ≤ j ≤ } where the cliques are defined by the adjacencies ( i, j ) ∼ ( i, ℓ ), 0 ≤ i ≤ ≤ j < ℓ ≤
3. In order to simplify the notation we will now denote vertices (0 , j ), (1 , j ) and(2 , j ) by Aj , Bj and Cj respectively. Ignoring the vertices Cj and the edges incident withthese vertices, the reduced graph is strongly (2 , , Aj and Bj also form an 8-cycle. Without loss of generality, wecan assume it to be ( A , B , A , B , A , B , A , B Aj and Cj form one of the following 8-cycles.(I) ( A , C , A , C , A , C , A , C A , C , A , C , A , C , A , C A , C , A , C , A , C , A , C A A B B B B Bj and Cj , which again must form an 8-cycle.Consider first case (I). It is not possible that Bj ∼ Cj for any j , since a 4-clique on the vertexset { A ( j + 1) , A ( j + 2) , Bj, Cj } would be created. This leaves just six possibilities.(i) ( B , C , B , C , B , C , B , C B , C , B , C , B , C , B , C B , C , B , C , B , C , B , C B , C , B , C , B , C , B , C B , C , B , C , B , C , B , C B , C , B , C , B , C , B , C B C B C B C B C A A A A B B B B C C C C , , A , B and C vertices in both possibility (i) and possibility (iv). In the lattercase the graph is antipodal; every pair of vertices is distance 1 or distance 2 apart, with theexception of the pairs { A , A } , { A , A } , { B , B } , { B , B } , { C , C } , { C , C } , which aredistance 3 apart. In the graph obtained from possibility (i), only the pairs { A , A } , { A , A } , { B , C } , { B , C } are distance 3 apart. 9ycle Clique formed(i) ( B , C , B , C , B , C , B , C B , C , B , C , B , C , B , C { A , B , B , C } (iii) ( B , C , B , C , B , C , B , C { A , B , B , C } (iv) ( B , C , B , C , B , C , B , C { A , B , B , C } (v) ( B , C , B , C , B , C , B , C { A , B , B , C } (vi) ( B , C , B , C , B , C , B , C { A , B , B , C } (vii) ( B , C , B , C , B , C , B , C B , C , B , C , B , C , B , C { A , B , B , C } (ix) ( B , C , B , C , B , C , B , C { A , B , C , C } (x) ( B , C , B , C , B , C , B , C { A , B , B , C } (xi) ( B , C , B , C , B , C , B , C { A , B , C , C } (xii) ( B , C , B , C , B , C , B , C B , C , B , C , B , C , B , C { A , B , C , C } (xiv) ( B , C , B , C , B , C , B , C { A , B , B , C } (xv) ( B , C , B , C , B , C , B , C { A , B , C , C } (xvi) ( B , C , B , C , B , C , B , C { A , B , B , C } (xvii) ( B , C , B , C , B , C , B , C { A , B , C , C } (xviii) ( B , C , B , C , B , C , B , C B , C , B , C , B , C , B , C { A , B , C , C } (xx) ( B , C , B , C , B , C , B , C { A , B , B , C } Table 1: Possible cycles for n = 3 , v = 4: Case (II)Alternatively we can use a computer package [2, 8] to determine the automorphism groupsof the two strongly (3 , A A B B B B C C C C
3) and ( A A B C B C B C B C B ∼ C { A , A , B , C } wouldbe created; nor that B ∼ C { A , A , B , C } would be created. This reducesthe number of possibilities for the 8-cycle on the vertices Bj and Cj to 20; 16 of which can alsobe eliminated because further 4-cliques are formed. In Table 1 we give a list of these possibilities,together with any cliques which occur.We find that the graphs obtained from case (II), possibilities (i), (vii), (xii) and (xviii) areisomorphic to the graph obtained from case (I), possibility (i) by the following permutations.Possibility Permutation(i) ( A B A B A B A B C C C A B C A B C A B C A B C A C B A C B A C B A C B A C A C A C A C B B B B n = 4 and v = 4, the original problem. An exhaustive consideration of themany cases along the lines of the proof of Theorem 2.10 is infeasible. However, by computersearch we are able to enumerate the non-isomorphic graphs; there are in fact six of them. InTable 2 we show these graphs by specifying the 8-cycles of edges between cliques in the formabove. Graph 1 in the table is Γ(4 , ,
4) is a circulant graph, as shown in Proposition 2.4. We may view it as a Cayley graph of Z with connection set {± , ± , ± , ± , } . The four 4-cliques arise from the elements {± , } ,and the edges between cliques from the other elements in the connection set.10 1 2 3 45678910111213 14 15Figure 7: The graph Γ(4 , v = 4 and n ≥
5, the enumeration becomes increasingly challenging. At n = 5 the computersearch yields exactly 24 graphs; at n = 6 there are at least 129, and at n = 7 at least 828.11raph Connections between cliques Automorphism group1 ( A , B , A , B , A , B , A , B D ( A , C , A , C , A , C , A , C B , C , B , C , B , C , B , C A , D , A , D , A , D , A , D B , D , B , D , B , D , B , D C , D , C , D , C , D , C , D A , B , A , B , A , B , A , B Z × Z ( A , C , A , C , A , C , A , C B , C , B , C , B , C , B , C A , D , A , D , A , D , A , D B , D , B , D , B , D , B , D C , D , C , D , C , D , C , D A , B , A , B , A , B , A , B Z ( A , C , A , C , A , C , A , C B , C , B , C , B , C , B , C A , D , A , D , A , D , A , D B , D , B , D , B , D , B , D C , D , C , D , C , D , C , D A , B , A , B , A , B , A , B Z × Z ( A , C , A , C , A , C , A , C B , C , B , C , B , C , B , C A , D , A , D , A , D , A , D B , D , B , D , B , D , B , D C , D , C , D , C , D , C , D A , B , A , B , A , B , A , B Z ( A , C , A , C , A , C , A , C B , C , B , C , B , C , B , C A , D , A , D , A , D , A , D B , D , B , D , B , D , B , D C , D , C , D , C , D , C , D A , B , A , B , A , B , A , B Z × Z ( A , C , A , C , A , C , A , C B , C , B , C , B , C , B , C A , D , A , D , A , D , A , D B , D , B , D , B , D , B , D C , D , C , D , C , D , C , D , Weakly clique-partitioned graphs
To answer Question 2, we again derive a simple upper bound based on counting edges. Recallthat in this version of the problem, we allow the possibility of additional v -cliques in our graphbut require that the decomposition into n vertex-disjoint v -cliques be unique. Lemma 3.1.
Let n ≥ and v ≥ . Let G be a weakly ( n, v ) -clique-partitioned graph. Then thenumber of edges in G is at most (cid:18) nv (cid:19) − n ( n − v . Proof.
To get an upper bound on the number of edges in G , we find a lower bound on thenumber of edges we must remove from a complete graph to make the weakly clique-partitionedproperty hold.We know G has nv vertices and can be partitioned into n vertex-disjoint v -cliques. Let N bethe number of edges removed from the complete graph K nv to obtain G .Since there are n ( n − / G , if N < n ( n − v/ X, Y of v -cliques in the decomposition which have fewer than v edgesmissing between them. So there is some x ∈ X connected to every vertex of Y , and some y ∈ Y connected to every vertex of X .This leads to a v -clique decomposition of G including new cliques ( X \ { x } ) ∪ { y } and ( Y \{ y } ) ∪ { x } , contrary to the uniqueness of the decomposition of G . So N ≥ n ( n − v/ G is (cid:18) nv (cid:19) − n ( n − v . Again we will call a weakly clique-partitioned graph attaining this bound maximal . To showthat maximal graphs exist, we define a new graph Γ ′ ( n, v ) with the same vertex set as before: { ( i, j ) : 0 ≤ i ≤ n − , ≤ j ≤ v − } .To define the edges of Γ ′ ( n, v ), we begin with the complete graph on this vertex set and removeall edges joining ( i,
0) to ( k, ℓ ) for all i = 0 , , . . . , n −
2, all k = i + 1 , . . . , n − ℓ = 0 , , . . . , v − ′ ( n, v ) is ( n − v + ( n − v + · · · + v = n ( n − v/ ′ ( n, v ), so eachset of v vertices of the form ( i, j ) for fixed i forms a v -clique, which we number i .This construction is illustrated in Figure 8 for the case n = 4 , v = 4. In Figure 8(a) we removeall edges from vertex (0 ,
0) to vertices in cliques 1 , ,
3. In Figure 8(b) we remove all edges fromvertex (1 ,
0) to vertices in cliques 2 , , Theorem 3.2.
Let n ≥ and v ≥ . Then Γ ′ ( n, v ) is weakly ( n, v ) -clique-partitioned and hencethe upper bound of Lemma 3.1 is attained.Proof. We proceed iteratively. Vertex (0 ,
0) is not adjacent to any vertex not in clique number0, so any v -clique decomposition must include clique 0.13 (a) (b) (c)Figure 8: Construction of the graph Γ ′ (4 ,
4) by deleting edgesIn the remainder of the graph, vertex (1 ,
0) is not adjacent to any vertex not in clique 1, so any v -clique decomposition must include clique 1 also.This argument can be repeated for each numbered v -clique, and so there is a unique decompo-sition of Γ ′ ( n, v ) into n v -cliques as required.In contrast with the strongly clique-partitioned case in Section 2, it turns out that these graphsare the unique weakly clique-partitioned graphs attaining the upper bound. In the case v = 2,weakly clique-partitioned graphs are precisely those which admit a unique perfect matching. Thestructure of edge-maximal graphs in this class was deduced by Hetyei [3]; see also [5, Corollary1.6]. By Hetyei’s results, the graphs Γ ′ ( n,
2) are the unique maximal weakly clique-partitionedgraphs.Our aim now is to extend this uniqueness result to all v ≥ n ≥
2. We begin with thecase n = 2. Theorem 3.3.
Let v ≥ . Let G be a maximal weakly (2 , v ) -clique-partitioned graph. Then G is isomorphic to Γ ′ (2 , v ) .Proof. G consists of two v -cliques X and Y with v ( v −
1) edges between them; in other wordsit is the complete graph K v with v edges missing between the two cliques.We show first that in one of the two cliques, each vertex is adjacent to precisely v − x ∈ X and y ∈ Y each adjacent to all verticesin the other clique; then ( X \ { x } ) ∪ { y } and its complement form a new v -clique partition of G . Without loss of generality, we may assume all vertices of Y are adjacent to precisely v − X . We now show that each vertex of Y must be adjacent to the same v − X .Let x ∈ X be a vertex not adjacent to all vertices in Y . Let M be the set of non-neighbours of x in Y , and let m = | M | . Since only v edges are missing between X and Y , there are at least m − X adjacent to all vertices in Y . Let S be a set of m − Y \ M ) ∪ { x } ∪ S and its complement are a v -clique decomposition of G . This decompositionis distinct from X and Y unless m = v .Thus there is one vertex in X not adjacent to any vertex in Y , and this accounts for all v missingedges between X and Y . Thus G is isomorphic to Γ ′ (2 , v ).It is immediate that if G is a weakly ( n, v )-clique-partitioned graph, then any m of the n v -cliques in the unique decomposition of G (1 ≤ m ≤ n ) induce a weakly ( m, v )-clique-partitionedsubgraph of G . The above discussion then shows that if X and Y are two of the v -cliques in amaximal graph, then exactly one of the following situations must occur.14a) There is a vertex x ∈ X not adjacent to any vertex in Y ;or(b) There is a vertex y ∈ Y not adjacent to any vertex in X .In the first situation we write X → Y ; otherwise Y → X . Where necessary, we will indicate thedistinguished vertex not adjacent to the other clique by the notation X ( x ) → Y or Y ( y ) → X asappropriate. Our first result towards the extension to n ≥ Lemma 3.4.
Let n ≥ and let v ≥ . Let G be a maximal weakly ( n, v ) -clique-partitionedgraph, and let X = { x , . . . , x v } , Y = { y , . . . , y v } and Z = { z , . . . , z v } be distinct cliques inthe decomposition of G . If X ( x i ) → Y and X ( x j ) → Z then i = j .Proof. The subgraph of S of G induced by X ∪ Y ∪ Z is weakly (3 , v )-clique partitioned. Sup-pose i = j . Up to isomorphism, we may assume i = 1, j = 2 and Y ( y ) → Z . Then thesets { x , z , . . . , z v } , { z , y , . . . , y v } and { y , x , . . . , x v } form another v -clique partition of S , acontradiction.In light of Lemma 3.4 we may drop the superscript notation and assume the vertices are num-bered such that if X → Y , then the distinguished vertex in X is x .If X and Y are v -cliques in the decomposition of G , then either X → Y or Y → X . The cliquesare therefore arranged in a tournament. The next step is to show that this tournament is acyclic. Lemma 3.5.
Let n ≥ and let v ≥ . Let G be a maximal weakly ( n, v ) -clique-partitionedgraph, and let X = { x , . . . , x v } , Y = { y , . . . , y v } and Z = { z , . . . , z v } be distinct cliques inthe decomposition of v . If X → Y and Y → Z then X → Z .Proof. If Z → X , then the sets { x , z , . . . , z v } , { y , x , . . . , x v } and { z , y , . . . , y v } form a v -clique decomposition of G .We are now ready to state the uniqueness result. Theorem 3.6.
Let n ≥ and let v ≥ . Let G be a maximal weakly ( n, v ) -clique-partitionedgraph. Then G is isomorphic to the graph Γ ′ ( n, v ) .Proof. The result follows from Theorem 3.3, Lemmas 3.4 and 3.5 and uniqueness of the acyclictournament (Lemma 2.8).The graphs Γ ′ ( n, v ) are of course far from regular. However, their structure is sufficiently tightlydefined to be able to compute their automorphism group. Proposition 3.7.
Let n ≥ and let v ≥ . Then Aut(Γ ′ ( n, v )) has order v ! (( v − n − andis isomorphic to S v × S n − v − .Proof. If X , X , . . . , X n are the n v -cliques in the composition of Γ ′ ( n, v ), then by a suitablelabelling they form a chain X → X → · · · → X n . The distinguished vertices in X i , 1 ≤ i ≤ n − i ( v − X i , 1 ≤ i ≤ n − i ( v −
1) + ( n − i ) v = nv − i . Any automorphism must clearly preserve the unique cliquedecomposition, but no cliques can be exchanged since the distinguished vertices all have differentvalency. The distinguished vertices are fixed by any automorphism, but the non-distinguishedvertices within any clique may be permuted freely. Clique X n has no distinguished vertex. Theresult follows. 15 Further links to extremal graph theory
Recall the following which is well-known. An n -vertex graph which does not contain any ( r + 1)-vertex clique may be constructed by partitioning the set of vertices into r parts of equal or nearlyequal size, and connecting two vertices by an edge whenever they belong to different parts. Thisis the Tur´an graph T ( n, r ). By “nearly equal size” is meant that the cardinality of any two partsdiffers by at most one. This graph has the largest number of edges among all K r +1 -free n -vertexgraphs [9, 10]. Putting r = v and replacing n by nv , the graph T ( nv, v ) is regular of valency n ( v −
1) and has n v ( v − / ′ ( n, v ),the unique maximal weakly ( n, v )-clique-partitioned graph on this parameter set. As observedin the previous section, the graphs Γ ′ ( n, v ) are not regular and arise in a different manner tothe graphs T ( nv, v ). But it is not without some interest that there exist two distinct families ofunique graphs on the same number of vertices with completely different properties.Probably of more interest though in this context are maximal strongly ( n, v )-clique-partitionedgraphs. Clearly these graphs contain no ( v +1)-clique. Let G be a maximal strongly ( n, v )-clique-partitioned graph and denote the number of edges by | G | . Then | G | = nv ( v − / nv ( n − v − / nv ( nv − n + 1) /
2. So | G | / | T ( nv, v ) | = ( nv − n + 1) /n ( v −
1) = 1 − ( n − /n ( v − → v → ∞ . Thus maximal strongly ( n, v )-clique-partitioned graphs form a family of graphsavoiding ( v + 1)-cliques, the number of edges of which approach asymptotically the number ofedges in the extremal graphs T ( nv, v ).Finally, we note the relationship between our questions and colouring problems. It is immediatethat the complement of a strongly ( n, v )-clique-partitioned graph is a graph with chromaticnumber n , and with the property that any proper n -colouring decomposes the graph into aunique set of colour classes, each of size v . It is shown in [4, Theorem 3.3] that the circulantgraph Cay( Z nv , {± , ± , . . . , ± ( n − } ) is an edge-minimal uniquely n -colourable graph of order nv . Since by [4, Theorem 2.2] the colour classes of such a Cayley graph must be cosets ofsome subgroup of order v in Z nv , it follows that this circulant graph is unique. Therefore itscomplement is the unique maximal strongly ( n, v )-clique partitioned circulant graph. Sinceby Proposition 2.4 our graph Γ( n, v ) is a circulant graph, it is therefore isomorphic to thecomplement of the graph Cay( Z nv , {± , ± , . . . , ± ( n − } ) from [4]. References [1] L. R. Foulds.
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