Cliques in graphs

Abstract

The main focus of this thesis is to evaluate kr(n, δ), the minimal number of r-cliques in graphs with n vertices and minimum degree δ. A fundamental result in Graph Theory states that a triangle-free graph of order n has at most n/4 edges. Hence, a triangle-free graph has minimum degree at most n/2, so if k3(n, δ) = 0 then δ ≤ n/2. For n/2 ≤ δ ≤ 4n/5, I have evaluated kr(n, δ) and determined the structures of the extremal graphs. For δ ≥ 4n/5, I give a conjecture on kr(n, δ), as well as the structures of these extremal graphs. Moreover, I have proved various partial results that support this conjecture. Let k r (n, δ) be the analogous version of kr(n, δ) for regular graphs. Notice that there exist n and δ such that kr(n, δ) = 0 but k reg r (n, δ) > 0. For example, a theorem of Andrasfai, Erdős and Sos states that any triangle-free graph of order n with minimum degree greater than 2n/5 must be bipartite. Hence k3(n, bn/2c) = 0 but k 3 (n, bn/2c) > 0 for n odd. I have evaluated the exact value k 3 (n, δ) for δ between 2n/5 + 12 √ n/5 and n/2 and determined the structure of these extremal graphs. At the end of the thesis, I investigate a question in Ramsey Theory. The Ramsey number Rk(G) of a graph G is the minimum number N , such that any edge colouring of KN with k colours contains a monochromatic copy of G. The constrained Ramsey number f(G, T ) of two graphs G and T is the minimum number N such that any edge colouring of KN with any number of colours contains a monochromatic copy of G or a rainbow copy of T . It turns out that these two quantities are closely related when T is a matching. Namely, for almost all graphs G, f(G, tK2) = Rt−1(G) for t ≥ 2.