Cliques in Graphs Excluding a Complete Graph Minor

This paper considers the following question: What is the maximum number of $k$-cliques in an $n$-vertex graph with no $K_t$-minor? This question generalises the extremal function for $K_t$-minors, which corresponds to the $k=2$ case. The exact answer is given for $t\leq 9$ and all values of $k$. We also determine the maximum total number of cliques in an $n$-vertex graph with no $K_t$-minor for $t\leq 9$. Several observations are made about the case of general $t$.


Introduction
A basic question of extremal graph theory asks: for a class G of graphs, what is the maximum number of edges in an n-vertex graph in G? The answer is called the extremal function for G. Consider the following two classical examples. Turán's Theorem [34] says that every n-vertex graph with no K t -subgraph has at most ( t−2 2t−2 )n 2 edges, with equality only for the complete (t − 1)-partite graph with n t−1 vertices in each colours class (called the Turán graph). And Euler's formula implies that the maximum number of edges in a planar graph with n ≥ 3 vertices equals 3n − 6.
One way to generalise these results is to consider cliques instead of edges. A clique in a graph is a set of pairwise adjacent vertices. A k-clique is a clique of cardinality k. Since a 2-clique is simply an edge, the following natural generalisations of the above question arise: For a class G of graphs, • what is the maximum number of k-cliques in an n-vertex graph in G, and • what is the maximum number of cliques in an n-vertex graph in G?
Zykov [36] generalised Turán's Theorem by answering the above questions for the class of graphs with no K t -subgraph. He proved that for t > k ≥ 0, every graph with n ≥ k vertices 15th November, 2015, revised 15th July 2016 * School of Mathematical Sciences, Monash University, Melbourne, Australia (david.wood@monash.edu).
Research supported by the Australian Research Council. and no K t -subgraph contains at most t−1 k ( n t−1 ) k cliques of size k, and every graph with n vertices and no K t -subgraph contains at most ( n t−1 + 1) t−1 cliques. Both bounds are tight for the Turán graph. Bounds on the number of k-cliques in graphs of given maximum degree have been extensively studied [1,6,12,17,18,35]. Several papers have established upper bounds on the number of k-cliques in terms of the number of vertices and the number of edges, or more generally, in terms of the number of (≤ k − 1)-cliques [5,10,11,16,29].
For planar graphs, Hakimi and Schmeichel [19] proved that the maximum number of triangles is 3n − 8, and Wood [35] proved that the maximum number of 4-cliques is n − 3, and in total the maximum number of cliques is 8n − 16. See [27] for earlier upper bounds for planar graphs and see [9] for an extension to arbitrary surfaces.
This paper considers these questions in graph classes defined by an excluded minor, thus generalising the above results for planar graphs. This direction has been recently pursued by several authors [13,14,23,26,28]. These works have focused on asymptotic results when the excluded minor is a general complete graph K t . The primary focus of this paper is exact results, when the excluded minor is K 3 , K 4 , K 5 , K 6 , K 7 , K 8 or K 9 (Sections 3-5). We also make several observations and conjectures about general K t -minor-free graphs (Sections 6-7).
While bounds on the number of cliques in minor-closed classes are of independent interest, such results have had diverse applications, including the asymptotic enumeration of minorclosed classes [26], and in the analysis of an algorithm for finding small separators [28], which in turn has been applied to finding shortest paths [31] and in matrix sparsification [2] for example.
Let cliques(n, t, k) be the maximum number of k-cliques in a K t -minor-free graph on n vertices. Let cliques(n, t) be the maximum number of cliques in a K t -minor-free graph on n vertices. Of course, if n ≤ t − 1 then K n is K t -minor-free, in which case cliques(n, t, k) = n k and cliques(n, t) = 2 n . (1) The following example provides an important lower bound on cliques(n, t, k). For an integer ℓ ≥ 1, an ℓ-tree is a graph defined recursively as follows. First, the complete graph K ℓ is an ℓ-tree. Then, if C is an ℓ-clique in an ℓ-tree, then the graph obtained by adding a new vertex adjacent only to C, is also a an ℓ-tree. Every ℓ-tree has tree-width at most ℓ, and thus contains no K ℓ+2 -minor. Observe that for every ℓ-tree G with n vertices, cliques(G, k) = ℓ k−1 (n − (ℓ+1)(k−1) k ) and cliques(G) = 2 ℓ (n − ℓ + 1). Hence for n ≥ t − 2 and t > k ≥ 1, The results of this paper show that these lower bounds hold with equality for many values of t and k. This is the case for n ∈ {t − 2, t − 1} by (1).  Mader [24] observed that Theorem 1 does not hold with t = 8. Let K c×2 be the complete c-partite graph K 2,2,...,2 with n = 2c vertices, which can be thought of as K 2c minus a perfect matching. In the t = 8 case, Theorem 1 would give a bound of 6n−21 on the number of edges in a K 8 -minor-free graph, whereas Mader [24] observed that K 2,2,2,2,2 has n = 10 vertices, 40 > 6 · 10 − 21 edges, and contains no K 8 -minor. K c×2 will be an important example throughout this paper. In general, Wood [35] proved that K c×2 contains no K t -minor where t = ⌊ 3 2 c⌋ + 1, and a K t−1 -minor in K c×2 is obtained from a K c subgraph by contracting a ⌊ c 2 ⌋-edge matching in the remaining graph. The following theorems summarise the main contributions of this paper.
except for (t, k) ∈ {(8, 2), (9, 2), (9, 3)} and certain values of n (made precise below) in which case Note that our proof depends on the case k = 2 and does not reprove the existing results.
We employ the following notation. For a vertex v in a graph G, let N (v) be the set of

Cockades
This section introduces a well known construction that will be important later. Let H be a graph containing a k-clique. An (H, k)-cockade is defined recursively as follows. First, H is an (H, k)-cockade. And if H 1 and H 2 are (H, k)-cockades, then the graph obtained from pasting H 1 and H 2 on a k-clique is an (H, k)-cockade. It is easy to count cliques in cockades.

Lemma 4.
For i ∈ {1, 2}, let G i be a graph with n i vertices such that cliques(G i , k) = a(n i − r) + r k , for some fixed a and r ≥ k. Let G be obtained by pasting G 1 and G 2 on an r-clique. Then G has n = n 1 + n 2 − r vertices, and cliques(G, k) = a(n − r) + r k . Proof.
Lemma 5. For i ∈ {1, 2}, let G i be a graph with n i vertices such that cliques(G i ) = a(n i − r) + 2 r , for some fixed a and r ≥ 0. Let G be obtained by pasting G 1 and G 2 on an r-clique. Then G has n = n 1 + n 2 − r vertices, and cliques(G) = a(n − r) + 2 r .
Lemma 6. For every (K c×2 , c)-cockade G on n vertices and for k ∈ {0, 1, . . . , c}, Proof. The first claim follows from Lemma 4 with r = c and a = 1 The second claim follows from Lemma 5 with r = c and a = 1

K t -minor-free graphs with t ≤ 7
This section proves Theorems 2 and 3 for t ≤ 7.
Theorem 7. For t ∈ {3, 4, 5, 6, 7} and k ∈ {1, 2, . . . , t − 1} and n ≥ t − 2, the maximum number of k-cliques in a K t -minor-free graph on n vertices satisfies Proof. The lower bound is provided by (2). For the upper bound, we proceed by induction on n + k. The claim is trivial if k = 1. Theorem 1 proves the claim when k = 2. Now assume that k ≥ 3. In the case n = t − 2 the claimed upper bound on cliques(n, t, k) is n k , which obviously holds. Let G be a K t -minor-free graph on n ≥ t − 1 vertices.
as desired.
Now assume that G has minimum degree at least t − 1. For each k-clique C in G send a charge of 1 k to each vertex in C. The charge received by each vertex v equals Theorem 8. For t ∈ {3, 4, 5, 6, 7}, the maximum number of cliques in a K t -minor-free graph on n ≥ t − 2 vertices equals 2 t−2 (n − t + 3).

Proof. The lower bound is (3). The upper bound follows from Theorem 7 since
Theorem 7 and 8 were previously proved for t = 5 by Wood [35] (using a different method).
We now prove Theorem 2 for K 8 -minor-free graphs with k ≥ 3.
Theorem 10. For k ∈ {3, 4, 5, 6, 7} the maximum number of k-cliques in a K 8 -minor-free graph on n ≥ 6 vertices satisfies Proof. The lower bound is provided by (2). For the upper bound, let G be a K 8 -minor-free graph on n ≥ 7 vertices. We proceed by induction on n with k fixed. In the base case with n ∈ {6, 7}, the result holds by (1). Now assume that n ≥ 8.
The remainder of the proof is analogous to the proof of Theorem 7, so we sketch it briefly. First, delete a vertex of degree at most 5 and apply induction. Now assume minimum degree at least 6. Charge each k-clique to its vertices, and count the charge at each vertex v with respect to deg(v) and the number of (k−1)-cliques in G(v), which is K 7 -minor-free (applying Theorem 7). Counting the total charge, (4) gives Since |E(G)| ≤ 6n − 21, it follows by the same analysis used in the proof of Theorem 7 that Theorem 11. The maximum number of cliques in a K 8 -minor-free graph on n ≥ 6 vertices equals 64(n − 5).
Essentially the same method used above determines the maximum number of k-cliques in a K 9 -minor-free graph for k ≥ 4.
Theorem 13. For k ∈ {4, 5, 6, 7, 8}, the maximum number of k-cliques in a K 9 -minor-free graph on n ≥ 7 vertices equals Proof. The lower bound is provided by (2). We proceed by induction on n with k fixed. In the base case with n ∈ {7, 8}, the result holds by (1). Let G be a K 9 -minor-free graph on n ≥ 9 vertices.
The remainder of the proof is analogous to the proof of Theorem 7, so we sketch it briefly. First, delete a vertex of degree at most 6 and apply induction. Now assume minimum degree at least 7. Charge each k-clique to its vertices, and count the charge at each vertex v with respect to deg(v) and the number of (k−1)-cliques in G(v), which is K 8 -minor-free (applying Theorem 10 since k − 1 ≥ 3). Counting the total charge, (4) gives Since |E(G)| ≤ 7n − 28, it follows by the same analysis used in the proof of Theorem 7 that .
With a bit more work, we now determine the maximum number of triangles in a K 9 -minor-free graph.
Lemma 15. Every 6-connected n-vertex m-edge K 9 -minor-free graph contains at most 4m − 7n triangles. Proof. First suppose G contains a vertex v with G(v) isomorphic to a (K 2,2,2,2,2 , 5)-cockade. If N (C) is a clique, then |N (C)| ≤ 5 (since no (K 2,2,2,2,2 , 5)-cockade contains K 6 ) and G is not 6-connected, which is a contradiction. Thus N (C) contains two non-adjacent vertices x and y. Let G ′ be obtained from G by contracting C to a vertex z and then contracting zx. Then xy is an edge of G ′ . Since every (K 2,2,2,2,2 , 5)-cockade is edge-maximal with no K 8 -minor, G ′ [N (v)] contains a K 8 -minor, and (with v) G ′ contains a K 9 -minor. Hence G contains a K 9 -minor, which is a contradiction. Now assume that G(v) is isomorphic to a (K 2,2,2,2,2 , 5)-cockade for no vertex v. Send a charge of 1 3 from each triangle to each of the three vertices in it. Each vertex v receives a charge equal to 1 3 |E(G(v))|, which is at most 2 deg(v) − 7 by Theorem 9 (which is applicable since deg(v) ≥ 6 and G(v) is K 8 -minor-free). The number of triangles, which equals the total charge, is at most v (2 deg(v) − 7) = 4m − 7n, as desired.
Theorem 16. The maximum number of cliques in a K 9 -minor-free graph on n ≥ 7 vertices equals 128(n − 6).
Proof. Let G be a K 9 -minor-free graph on n ≥ 7 vertices.

Total Number of Cliques in K t -minor-free Graphs
This section considers the total number of cliques in K t -minor-free graphs for arbitrary t.
Recall that cliques(n, t) is the maximum number of cliques in a K t -minor-free graph on n vertices. The best lower bound on cliques(n, t) is due to Wood [35], who observed that K c×2 contains no K t -minor where t = ⌊ 3 2 c⌋ + 1. Thus Upper bounds on cliques(n, t) have been intensely studied over the past ten years, culminating in the recent upper bound by Fox and Wei [14] that matches the lower bound in (5) up to a lower order term. These results are summarised in the following Note that several authors have also studied the maximum number of cliques in graphs excluding a given subdivision [15,23] or immersion [15].
The remainder of this section considers the following question: what is the maximum integer t 0 such that cliques(n, t) = 2 t−2 (n − t + 3) for all t ≤ t 0 (thus matching the lower bound in (3))? Theorem 3 shows that t 0 ≥ 9.
Given that K c×2 provides an essentially tight lower bound on cliques(n, t), we now examine complete multipartite graphs in more detail. Consider a complete c-partite graph G = K n 1 ,...,nc where n 1 ≥ · · · ≥ n c ≥ 1 and c ≥ 2. Then n = c i=1 n i is the number of vertices. Wood [35] proved that G contains no K t -minor, where t = min 1 2 (n + c) + 1, n − n 1 + 2 , and a K t−1 -minor in G can be obtained from a K c subgraph by contracting a maximum matching in the remaining graph. If ⌊ 1 2 (n + c)⌋ ≤ n − n 1 + 1 then we say G is balanced, otherwise G is unbalanced (in which case the largest colour class is 'very' large). First suppose that G is unbalanced. Let m := n−n 1 c−1 be the average size of a colour class except the largest colour class. Then m ≥ 1 and m + 1 ≤ 2 m . Thus That is, every unbalanced complete multipartite graph satisfies the bound. Now consider the case in which G is balanced. Let m := n c be the average size of a colour class. Then t − 2 ≥ 1 2 (n + c − 1) − 1 = 1 2 (c(m + 1) − 3) and n − t + 3 ≥ n − 1 2 (n + c) + 3 = 1 2 (n − c) + 3 = 1 2 c(m − 1) + 3. Assume that m ≥ 3. Then (m + 1) 2 ≤ 2 m+1 and (m + 1) c ≤ 2 c(m+1)/2 . Since 1 2 c(m − 1) + 3 > 2 3/2 , That is, balanced complete multipartite graphs with an average of at least three vertices per colour class satisfy the bound. Thus, if a complete multipartite graph has more than 2 t−2 (n − t + 3) cliques, then it is balanced and has an average of less than three vertices per colour class. This is why K c×2 is a critical example. Computer search establishes that for t ≤ 49 every such complete multipartite graph has at most 2 t−2 (n − t + 3) cliques, but for t ≥ 50 there is a value of c such that K c×2 or K 1,c×2 or K 1,1,c×2 has no K t -minor and contains more than 2 t−2 (n − t + 3) cliques. Indeed K c×2 satisfies this property for t ≥ 62. We therefore make the following conjecture.
A graph is d-degenerate if every subgraph has minimum degree at most d. Wood [35] determined the maximum total number of cliques in a d-degenerate graph. Essentially the same proof determines the maximum number of k-cliques.

Lemma 18.
For every d-degenerate graph G with n ≥ d + 1 vertices,

Proof.
We proceed by induction on n. For the base case with n = d + 1, the number of k-cliques is at most n k = d k−1 (n − k−1 k (d + 1)). Let G be a d-degenerate graph with n ≥ d + 2 vertices. There is a vertex v of degree at most d in G. The number of k-cliques In total, the number of k-cliques is at most d k−1 (n − k−1 k (d + 1)) .
The bound in Lemma 18 is tight for d-trees. The above-mentioned results of Kostochka [21,22] and Thomason [32,33] show that K t -minor-free graphs are ct √ log t-degenerate for some constant c. Lemma 18 thus implies For fixed k, this bound is tight up to a constant factor as we now explain. Bollobás et al. [3] proved that for a suitable constant c > 0 and for large t, a random graph on n = ct √ log t vertices has no K t -minor with high probability. Here each edge is chosen independently with probability 1 2 . Thus, the expected number of k-cliques is n k /2 ( k 2 ) . It follows that for large t, there exists an n-vertex graph with no K t -minor and with at least n k /2 ( k 2 ) k-cliques. Note that n k /2 ( k 2 ) = c ′ (t √ log t) k−1 n for a suitable constant c ′ . Thus there exists an n-vertex graph G with no K t -minor such that cliques(G, k) ≥ c ′ (t √ log t) k−1 n. Taking disjoint copies of G gives a graph with the same property, where n ≫ t. Summarising, for fixed k, there are constants c 1 and c 2 such that c 1 (t log t) k−1 n ≤ cliques(n, t, k) ≤ c 2 (t log t) k−1 n.
Thus cliques(n, t, k) is determined up to a constant factor for fixed k. But as k increases with t, determining cliques(n, t, k) is wide open. First note that a random graph will have few large cliques. In fact, the size of the largest clique in a random graph on t vertices is sharply concentrated around 2 log 2 t [4,25]. This motivates the following conjecture about 'large' cliques in K t -minor-free graphs.
Thus K c×2 satisfies Conjecture 20 for a particular value of k if and only if (6) holds. We now show that (6) is not satisfied by K c×2 for small k; that is, K c×2 has many small cliques.