Maximizing the number of cliques in graphs with given matching number

Abstract

A matching M in a graph G is a set of disjoint edges of G. The matching number ν(G) is the number of edges in a maximum matching of G. The number of cliques of size s in G is denoted by Ns(G). Given positive integers n, k, and r, the graph H(n,k,r) is the one obtained from K2k+1−r by connecting each vertex of an independent set of size n−(2k+1−r) to the same r vertices chosen from K2k+1−r. Set hs(n,k,r)=Ns(H(n,k,r))=2k+1−rs+(n−2k−1+r)rs−1. In 1959, Erdős and Gallai proved that if G is a graph with n≥2k+2 vertices and ν(G)≤k, then e(G)≤max{2k+12,h2(n,k,k)} and characterized the extremal graphs. In this paper, motivated by a question proposed by Füredi, Kostochka, and Luo, we first extend Erdős and Gallai’s result. To be precisely, we prove that if G is a graph with n≥2k+2 vertices, minimum degree δ, and ν(G)≤k, then Ns(G)≤max{hs(n,k,δ),hs(n,k,k)} for each s≥2. Moreover, we are able to prove stability versions of this result specialized to edges. Namely, we prove if e(G)>max{h2(n,k,δ),h2(n,k,k−2)} and n≥2k+2, then G is a subgraph of H(n,k,k) or H(n,k,k−1). If e(G)>max{h2(n,k,δ+2),h2(n,k,k)} and n≥2k+11, then G is a subgraph of H(n,k,δ) or H(n,k,δ+1).