We consider the structure of Kr-free graphs with large minimum degree, and show that such graphs with minimum degree δ > (2r − 5)n/(2r − 3) are homomorphic to the join Kr−3 ∨ H where H is a triangle-free graph. In particular this allows us to generalize results from triangle-free graphs and show that Kr-free graphs with such minimum degree have chromatic number at most r+ 1. We also consider th...

(Throughout this section, log denotes base-2 logarithm.) We now sketch a proof of this theorem. Proof: Let Xk be the number of k-cliques in a graph G drawn from Gn, 2 . By linearity of expectation we have g(k) := E[Xk] = ( n k ) 2−( k 2) and let us define k0(n) to be the largest value of k such that g(k) ≥ 1. An easy calculation (Exercise!) shows that k0(n) ∼ 2 log n. (To see this is plausible,...

The clique number C of a graph is the largest clique size in the graph. For a random geometric graph of n vertices, taken uniformly at random, including an edge beween two vertices if their distance, taken with the uniform norm, is less than a parameter r on a torus Ta, we find the asymptotic behaviour of the clique number. Setting θ = ( r a ), in the subcritical regime where θ = o( 1 n ), we e...

A clique colouring of a graph is a colouring of the vertices such that no maximal clique is monochromatic (ignoring isolated vertices). The least number of colours in such a colouring is the clique chromatic number. Given n points x1, . . . ,xn in the plane, and a threshold r > 0, the corresponding geometric graph has vertex set {v1, . . . , vn}, and distinct vi and vj are adjacent when the Euc...

In this paper we study the edge-thickness and the clique-thickness of a graph. The edge-thickness of a graph is defined as the thickness of the family of edges. The clique-thickness of a graph is defined as the thickness of the family of maximal cliques. Edges and maximal cliques of a graph are both considered as a collection of subsets of the vertex set. On one hand, we introduce a new paramet...

Many computer vision and patter recognition problems are intimately related to the maximum clique problem. Due to the intractability of this problem, besides the development of heuristics, a research direction consists in trying to find good bounds on the clique number of graphs. This paper introduces a new spectral upper bound on the clique number of graphs, which is obtained by exploiting an ...