The maximum number of cliques in a graph embedded in a surface
Vida Dujmović, Gašper Fijavž, Gwenaël Joret, Thom Sulanke, David R. Wood
OON THE MAXIMUM NUMBER OF CLIQUESIN A GRAPH EMBEDDED IN A SURFACE
VIDA DUJMOVI ´C, GAˇSPER FIJAVˇZ, GWENA¨EL JORET, THOM SULANKE, AND DAVID R. WOOD
Abstract.
This paper studies the following question: Given a surface Σ and an integer n , what isthe maximum number of cliques in an n -vertex graph embeddable in Σ? We characterise the extremalgraphs for this question, and prove that the answer is between 8( n − ω ) + 2 ω and 8 n + ω + o (2 ω ),where ω is the maximum integer such that the complete graph K ω embeds in Σ. For the surfaces S , S , S , N , N , N and N we establish an exact answer. MSC Classification : 05C10 (topological graph theory), 05C35 (extremal problems) Introduction A clique in a graph is a set of pairwise adjacent vertices. Let c ( G ) be the number of cliques in agraph G . For example, every set of vertices in the complete graph K n is a clique, and c ( K n ) = 2 n .This paper studies the following question at the intersection of topological and extremal graphtheory: Given a surface Σ and an integer n , what is the maximum number of cliques in an n -vertexgraph embeddable in Σ?For previous bounds on the maximum number of cliques in certain graph families see [5, 6, 13,14, 22, 23] for example. For background on graphs embedded in surfaces see [11, 21]. Every surfaceis homeomorphic to S g , the orientable surface with g handles, or to N h , the non-orientable surfacewith h crosscaps. The Euler characteristic of S g is 2 − g . The Euler characteristic of N h is 2 − h .The orientable genus of a graph G is the minimum integer g such that G embeds in S g . The non-orientable genus of a graph G is the minimum integer h such that G embeds in N h . The orientablegenus of K n ( n ≥
3) is (cid:100) ( n − n − (cid:101) , and its non-orientable genus is (cid:100) ( n − n − (cid:101) , exceptthat the non-orientable genus of K is 3. Date : October 24, 2018.This work was supported in part by the Actions de Recherche Concert´ees (ARC) fund of the Communaut´efran¸caise de Belgique. Vida Dujmovi´c is supported by the Natural Sciences and Engineering Research Council ofCanada. Gaˇsper Fijavˇz is supported in part by the Slovenian Research Agency, Research Program P1-0297. Gwena¨elJoret is a Postdoctoral Researcher of the Fonds National de la Recherche Scientifique (F.R.S.–FNRS). David Woodis supported by a QEII Research Fellowship from the Australian Research Council. We consider simple, finite, undirected graphs G with vertex set V ( G ) and edge set E ( G ). A K subgraph of G iscalled a triangle of G . For background graph theory see [4]. a r X i v : . [ m a t h . C O ] M a r VIDA DUJMOVI´C, GAˇSPER FIJAVˇZ, GWENA¨EL JORET, THOM SULANKE, AND DAVID R. WOOD
Throughout the paper, fix a surface Σ with Euler characteristic χ . If Σ = S then let ω = 3,otherwise let ω be the maximum integer such that K ω embeds in Σ. Thus ω = (cid:98) (7 + √ − χ ) (cid:99) except for Σ = S and Σ = N , in which case ω = 3 and ω = 6, respectively.To avoid trivial exceptions, we implicitly assume that | V ( G ) | ≥ S .Our first main result is to characterise the n -vertex graphs embeddable in Σ with the maximumnumber of cliques; see Theorem 1 in Section 2. Using this result we determine an exact formulafor the maximum number of cliques in an n -vertex graph embeddable in each of the the sphere S ,the torus S , the double torus S , the projective plane N , the Klein bottle N , as well as N and N ; see Section 3. Our third main result estimates the maximum number of cliques in terms of ω .We prove that the maximum number of cliques in an n -vertex graph embeddable in Σ is between8( n − ω ) + 2 ω and 8 n + ω + o (2 ω ); see Theorem 2 in Section 4.2. Characterisation of Extremal Graphs
The upper bounds proved in this paper are of the form: every graph G embeddable in Σ satisfies c ( G ) ≤ | V ( G ) | + f (Σ) for some function f . Define the excess of G to be c ( G ) − | V ( G ) | . Thus theexcess of G is at most Q if and only if c ( G ) ≤ | V ( G ) | + Q . Theorem 2 proves that the maximumexcess of a graph embeddable in Σ is finite.In this section we characterise the graphs embeddable in Σ with maximum excess. A triangulation of Σ is an embedding of a graph in Σ in which each facial walk has three vertices and three edgeswith no repetitions. (We assume that every face of a graph embedding is homeomorphic to a disc.) Lemma 1.
Every graph G embeddable in Σ with maximum excess is a triangulation of Σ .Proof. Since adding edges within a face increases the number of cliques, the vertices on the boundaryof each face of G form a clique.Suppose that some face f of G has at least four distinct vertices in its boundary. Let G (cid:48) be thegraph obtained from G by adding one new vertex adjacent to four distinct vertices of f . Thus G (cid:48) isembeddable in Σ, has | V ( G ) | + 1 vertices, and has c ( G ) + 16 cliques, which contradicts the choiceof G . Now assume that every face of G has at most three distinct vertices.Suppose that some face f of G has repeated vertices. Thus the facial walk of f contains vertices u, v, w, v in this order (where v is repeated in f ). Let G (cid:48) be the graph obtained from G by addingtwo new vertices p and q , where p is adjacent to { u, v, w, q } , and q is adjacent to { u, v, w, p } . So G (cid:48) is embeddable in Σ and has | V ( G ) | + 2 vertices. If S ⊆ { p, q } and S (cid:54) = ∅ and T ⊆ { u, v, w } , then S ∪ T is a clique of G (cid:48) but not of G . It follows that G (cid:48) has c ( G ) + 24 cliques, which contradicts thechoice of G . Hence no face of G has repeated vertices, and G is a triangulation of Σ. (cid:3) N THE MAXIMUM NUMBER OF CLIQUES IN A GRAPH EMBEDDED IN A SURFACE 3
Let G be a triangulation of Σ. An edge vw of G is reducible if vw is in exactly two triangles in G .We say G is irreducible if no edge of G is reducible [2, 3, 7, 9, 10, 12, 17, 19, 20]. Note that K is atriangulation of S , and by the above definition, K is irreducible. In fact, it is the only irreducibletriangulation of S . We take this somewhat non-standard approach so that Theorem 1 below holdsfor all surfaces.Let vw be a reducible edge of a triangulation G of Σ. Let vwx and vwy be the two faces incidentto vw in G . As illustrated in Figure 1, let G/vw be the graph obtained from G by contracting vw ;that is, delete the edges vw, wy, wx , and identify v and w into v . G/vw is a simple graph since x and y are the only common neighbours of v and w . Indeed, G/vw is a triangulation of Σ. Conversely,we say that G is obtained from G/vw by splitting the path xvy at v . If, in addition, xy ∈ E ( G ),then we say that G is obtained from G/vw by splitting the triangle xvy at v . Note that xvy neednot be a face of G/vw . In the case that xvy is a face, splitting xvy is equivalent to adding a newvertex adjacent to each of x, v, y . v wyxG vyx G/vw contractionsplitting Figure 1.
Contracting a reducible edge.Graphs embeddable in Σ with maximum excess are characterised in terms of irreducible triangu-lations as follows.
Theorem 1.
Let Q be the maximum excess of an irreducible triangulation of Σ . Let X be the setof irreducible triangulations of Σ with excess Q . Then the excess of every graph G embeddable in Σ is at most Q , with equality if and only if G is obtained from some graph in X by repeatedly splittingtriangles.Proof. We proceed by induction on | V ( G ) | . By Lemma 1, we may assume that G is a triangulationof Σ. If G is irreducible, then the claim follows from the definition of X and Q . Otherwise, someedge vw of G is in exactly two triangles vwx and vwy . By induction, the excess of G/vw is at most Q , with equality if and only if G/vw is obtained from some H ∈ X by repeatedly splitting triangles.Hence c ( G/vw ) ≤ | V ( G/vw ) | + Q . VIDA DUJMOVI´C, GAˇSPER FIJAVˇZ, GWENA¨EL JORET, THOM SULANKE, AND DAVID R. WOOD
Observe that every clique of G that is not in G/vw is in { A ∪ { w } : A ⊆ { x, v, y }} . Thus c ( G ) ≤ c ( G/vw ) + 8, with equality if and only if xvy is a triangle. Hence c ( G ) ≤ | V ( G ) | + Q ; thatis, the excess of G is at most Q .Now suppose that the excess of G equals Q . Then the excess of G/vw equals Q , and c ( G ) = c ( G/vw ) + 8 (implying xvy is a triangle). By induction,
G/vw is obtained from H by repeatedlysplitting triangles. Therefore G is obtained from H by repeatedly splitting triangles.Conversely, suppose that G is obtained from some H ∈ X by repeatedly splitting triangles. Then xvy is a triangle and G/vw is obtained from H by repeatedly splitting triangles. By induction, theexcess of G/vw equals Q , implying the excess of G equals Q . (cid:3) Low-Genus Surfaces
To prove an upper bound on the number of cliques in a graph embedded in Σ, by Theorem 1,it suffices to consider irreducible triangulations of Σ with maximum excess. The complete list ofirreducible triangulations is known for S , S , S , N , N , N and N . In particular, Steinitz andRademacher [16] proved that K is the only irreducible triangulation of S (under our definition ofirreducible). Lavrenchenko [9] proved that there are 21 irreducible triangulations of S , each withbetween 7 and 10 vertices. Sulanke [17] proved that there are 396 ,
784 irreducible triangulationsof S , each with between 10 and 17 vertices. Barnette [1] proved that the embeddings of K and K − K in N are the only irreducible triangulations of N . Sulanke [20] proved that thereare 29 irreducible triangulations of N , each with between 8 and 11 vertices (correcting an earlierresult by Lawrencenko and Negami [10]). Sulanke [17] proved that there are 9708 irreducibletriangulations of N , each with between 9 and 16 vertices. Sulanke [17] proved that there are6 , ,
982 irreducible triangulations of N , each with between 9 and 22 vertices. Using the lists ofall irreducible triangulations due to Sulanke [18] and a naive algorithm for counting cliques , wehave computed the set X in Theorem 1 for each of the above surfaces; see Table 1. This data withTheorem 1 implies the following results. Proposition 1.
Every planar graph G with | V ( G ) | ≥ | V ( G ) | −
16 cliques, as provedby Wood [22]. Moreover, a planar graph G has 8 | V ( G ) | −
16 cliques if and only if G is obtainedfrom the embedding of K in S by repeatedly splitting triangles. Proposition 2.
Every toroidal graph G has at most 8 | V ( G ) | + 72 cliques. Moreover, a toroidalgraph G has 8 | V ( G ) | + 72 cliques if and only if G is obtained from the embedding of K in S byrepeatedly splitting triangles (see Figure 2). The code is available from the authors upon request.
N THE MAXIMUM NUMBER OF CLIQUES IN A GRAPH EMBEDDED IN A SURFACE 5
Table 1.
The maximum excess of an n -vertex irreducible triangulation of Σ.Σ χ ω n = 3 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 max S − − S
48 40 32 72 S −
160 136 128 120 96 88 80 208 N N
48 48
40 32 48 N −
104 104
96 80 80 72 64 56 104 N −
208 152 136 136 136 128 120 112 107 99 91 83 75 216
Figure 2. K embedded in the torus, and K embedded in the projective plane. Proposition 3.
Every graph G embeddable in S has at most 8 | V ( G ) | + 208 cliques. Moreover, agraph G embeddable in S has 8 | V ( G ) | + 208 cliques if and only if G is obtained from one of thefollowing two graph embeddings in S by splitting repeatedly triangles :graph Proposition 4.
Every projective planar graph G has at most 8 | V ( G ) | + 16 cliques. Moreover, aprojective planar graph G has 8 | V ( G ) | + 16 cliques if and only if G is obtained from the embeddingof K in N by repeatedly splitting triangles (see Figure 2). Proposition 5.
Every graph G embeddable in the Klein bottle N has at most 8 | V ( G ) | +48 cliques.Moreover, a graph G embeddable in N has 8 | V ( G ) | + 48 cliques if and only G is obtained from oneof the following three graph embeddings in N by repeatedly splitting triangles (see Figure 3):graph This representation describes a graph with vertex set { a, b, c, . . . } by adjacency lists of the vertices in order a, b, c, . . . . The graph VIDA DUJMOVI´C, GAˇSPER FIJAVˇZ, GWENA¨EL JORET, THOM SULANKE, AND DAVID R. WOOD graph hh hh abb ccd de efgg ee ee aa bccdfg ghh cc cc aab bdeef fg hh i
Figure 3.
Irreducible triangulations of N with maximum excess: left-to-right Proposition 6.
Every graph G embeddable in N has at most 8 | V ( G ) | + 104 cliques. Moreover,a graph G embeddable in N has 8 | V ( G ) | + 104 cliques if and only G is obtained from one of thefollowing 15 graph embeddings in N by repeatedly splitting triangles:graph N THE MAXIMUM NUMBER OF CLIQUES IN A GRAPH EMBEDDED IN A SURFACE 7
Proposition 7.
Every graph G embeddable in N has at most 8 | V ( G ) | + 216 cliques. Moreover, agraph G embeddable in N has 8 | V ( G ) | + 216 cliques if and only if G is obtained from one of thefollowing three graph embeddings in N by repeatedly splitting triangles:graph A Bound for all Surfaces
Recall that Σ is a surface with Euler characteristic χ , and if Σ = S then ω = 3, otherwise ω isthe maximum integer such that K ω embeds in Σ. We start with the following upper bound on theminimum degree of a graph. Lemma 2.
Assume Σ (cid:54) = S . Then every graph G embeddable in Σ has minimum degree at most ω − ω − | V ( G ) | . Proof.
By the definition of ω , the complete graph K ω +1 cannot be embedded in Σ. Thus if Σ = S g then g = (2 − χ ) ≤ (cid:100) ( ω − ω − (cid:101) −
1, and if Σ = N h then h = 2 − χ ≤ (cid:100) ( ω − ω − (cid:101) − − χ ≤ ( ω − ω − − . That is,(1) − χ ≤ ω − ω − . Say G has minimum degree d . It follows from Euler’s Formula that | E ( G ) | ≤ | V ( G ) | − χ . By(1), d ≤ | E ( G ) || V ( G ) | ≤ | V ( G ) | − χ | V ( G ) | ≤ ω − ω − | V ( G ) | . (cid:3) For graphs in which the number of vertices is slightly more than ω , Lemma 2 can be reinterpretedas follows. Lemma 3.
Assume Σ (cid:54) = S . Let s := (cid:100)√ ω + 11 − (cid:101) ≥ . Let G be a graph embeddable in Σ . If G has at most ω + 1 vertices, then G has minimum degree at most ω − . If G has at least ω + j vertices, where j ∈ [2 , s ] , then G has minimum degree at most ω − j + 1 .Proof. Say G has minimum degree d . If | V ( G ) | ≤ ω , then trivially d ≤ ω −
1. If | V ( G ) | = ω + 1,then G is not complete (by the definition of ω ), again implying that d ≤ ω −
1. Now assume
VIDA DUJMOVI´C, GAˇSPER FIJAVˇZ, GWENA¨EL JORET, THOM SULANKE, AND DAVID R. WOOD | V ( G ) | ≥ ω + j for some j ∈ [2 , s ]. By Lemma 2, d ≤ ω − ω − ω + j = ω − j + 1 + j + 5 j − ω + j . Since j ≤ s < √ ω + 11 −
2, we have j + 5 j − ≤ s + 4 s − j < ω + j . It follows that d ≤ ω − j + 1. (cid:3) Now we prove our first upper bound on the number of cliques.
Lemma 4.
Assume Σ (cid:54) = S . Let s := (cid:100)√ ω + 11 − (cid:101) ≥ . Let G be an n -vertex graph embeddablein Σ . Then c ( G ) ≤ ω if n ≤ ω + s, ω + ( n − ω − s )2 ω − s +1 otherwise.Proof. Let v , v , . . . , v n be an ordering of the vertices of G such that v i has minimum degree in thesubgraph G i := G − { v , . . . , v i − } . Let d i be the degree of v i in G i (which equals the minimumdegree of G i ). Charge each non-empty clique C in G to the vertex v i ∈ C with i minimum. Chargethe clique ∅ to v n .We distinguish three types of vertices. Vertex v i is type-1 if i ∈ [1 , n − ω − s ]. Vertex v i is type-2if i ∈ [ n − ω − s + 1 , n − ω ]. Vertex v i is type-3 if i ∈ [ n − ω + 1 , n ].Each clique charged to a type-3 vertex is contained in { v n − ω +1 , . . . , v n } , and there are at most2 ω such cliques.Say C is a clique charged to a type-1 or type-2 vertex v i . Then C − { v i } is contained in N G i ( v i ),which consists of d i vertices. Thus the number of cliques charged to v i is at most 2 d i . Recall that d i equals the minimum degree of G i , which has n − i + 1 vertices.If v i is type-2 then, by Lemma 3 with j = n − ω − i + 1 ∈ [1 , s ], we have d i ≤ ω − j + 1, and d i ≤ ω − j if j = 1. Thus the number of cliques charged to type-2 vertices is at most2 ω − + s (cid:88) j =2 ω − j +1 ≤ ω − + ω − (cid:88) j =1 j <
32 2 ω . If v i is type-1 then G i has more than ω + s vertices, and thus d i ≤ ω − s + 1 by Lemma 3 with j = s . Thus the number of cliques charged to type-1 vertices is at most ( n − ω − s )2 ω − s +1 . (cid:3) We now prove the main result of this section; it provides lower and upper bound on the maximumnumber of cliques in a graph embeddable in Σ.
Theorem 2.
Every n -vertex graph embeddable in Σ contains at most n + ω + o (2 ω ) cliques.Moreover, for each n ≥ ω , there is an n -vertex graph embeddable in Σ with n − ω ) + 2 ω cliques. N THE MAXIMUM NUMBER OF CLIQUES IN A GRAPH EMBEDDED IN A SURFACE 9
Proof.
To prove the upper bound, we may assume that Σ (cid:54) = S , and by Theorem 1, we need onlyconsider n -vertex irreducible triangulations of Σ. Joret and Wood [7] proved that, in this case, n ≤ − χ . By (1), n ≤ − χ ≤
22 + ( ω − ω − < ω . If n ≤ ω + s then c ( G ) ≤ ω by Lemma 4. If n > ω + s then by the same lemma, c ( G ) ≤ ω + (3 ω − ω − s )2 ω − s +1 < ω + 3 ω ω − s +1 < ω + 2 ω − s +2 log ω +3 . Since s ∈ Θ( √ ω ), we have c ( G ) ≤ ω + o (2 ω ).To prove the lower bound, start with K ω embedded in Σ (which has 2 ω cliques). Now, whilethere are less than n vertices, insert a new vertex adjacent to each vertex of a single face. Each newvertex adds at least 8 new cliques. Thus we obtain an n -vertex graph embedded in Σ with at least8( n − ω ) + 2 ω cliques. (cid:3) Concluding Conjectures
We conjecture that the upper bound in Theorem 2 can be improved to more closely match thelower bound.
Conjecture 1.
Every graph G embeddable in Σ has at most 8 | V ( G ) | + 2 ω + o (2 ω ) cliques.If K ω triangulates Σ, then we conjecture the following exact answer. Conjecture 2.
Suppose that K ω triangulates Σ. Then every graph G embeddable in Σ has atmost 8( | V ( G ) | − ω ) + 2 ω cliques, with equality if and only if G is obtained from K ω by repeatedlysplitting triangles.By Theorem 1, this conjecture is equivalent to: Conjecture 3.
Suppose that K ω triangulates Σ. Then K ω is the only irreducible triangulation ofΣ with maximum excess.The results in Section 3 confirm Conjectures 2 and 3 for S , S and N .Now consider surfaces possibly with no complete graph triangulation. Then the bound c ( G ) ≤ | V ( G ) | − ω ) + 2 ω (in Conjecture 2) is false for S , N , N and N . Loosely speaking, this isbecause these surfaces have ‘small’ ω compared to χ . In particular, ω = (cid:98) (7 + √ − χ ) (cid:99) exceptfor S and N , and ω = (7 + √ − χ ) if and only if K ω triangulates Σ (cid:54) = S . This phenomenonmotivates the following conjecture: Conjecture 4.
Every graph G embeddable in Σ has at most8 | V ( G ) | − (cid:112) − χ ) + 2 (7+ √ − χ ) / cliques, with equality if and only if K ω triangulates Σ and G is obtained from K ω by repeatedlysplitting triangles.There are two irreducible triangulations of S with maximum excess, there are three irreducibletriangulations of N with maximum excess, there are 15 irreducible triangulations of N with max-imum excess, and there are three irreducible triangulations of N with maximum excess. Thissuggests that for surfaces with no complete graph triangulation, a succinct characterisation of theextremal examples (as in Conjecture 3) might be difficult. Nevertheless, we conjecture the followingstrengthening of Conjecture 3 for all surfaces. Conjecture 5.
Every irreducible triangulation of Σ with maximum excess contains K ω as a sub-graph.A triangulation of a surface Σ is vertex-minimal if it has the minimum number of vertices in atriangulation of Σ. Of course, every vertex-minimal triangulation is irreducible. Ringel [15] andJungerman and Ringel [8] together proved that the order of a vertex-minimal triangulation is ω if K ω triangulates Σ, is ω + 2 if Σ ∈ { S , N , N } , and is ω + 1 for every other surface.Triangulations N and N are the only triangulations in Propositions 1–7 thatare not vertex minimal. Triangulation N is obtained from two embeddings of K in N joined at the face bdf (see Figure 3). Triangulation N is obtained by joining an embeddingof K in N and an embedding of K in S at the face bdf .Every other triangulation in Propositions 1–7 is obtained from an embedding of K ω by adding(at most two) vertices and edges until a vertex minimal triangulation is obtained. This providessome evidence for our final conjecture: Conjecture 6.
For every surface Σ, the maximum excess is attained by some vertex-minimaltriangulation of Σ that contains K ω as a subgraph. Moreover, if Σ (cid:54)∈ { N , N } then every irreducibletriangulation with maximum excess is vertex-minimal and contains K ω as a subgraph.We have verified Conjectures 4, 5 and 6 for S , S , S , N , N , N and N . References [1]
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N THE MAXIMUM NUMBER OF CLIQUES IN A GRAPH EMBEDDED IN A SURFACE 13
School of Computer ScienceCarleton UniversityOttawa, Canada
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