Tight Bounds Towards a Conjecture of Gallai

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 number, then contains at most $$n-k+3$$ copies a clique size $$k-1$$ . This answers problem Abbott and Zhou provides tight bound on conjecture Gallai.