Tight Bounds Towards a Conjecture of Gallai

Abstract

We prove that for $$n>k\ge 3$$ , if G is an n-vertex graph with chromatic number k but any of its proper subgraphs has smaller chromatic number, then G contains at most $$n-k+3$$ copies of a clique of size $$k-1$$ . This answers a problem of Abbott and Zhou and provides a tight bound on a conjecture of Gallai.