Large cliques in graphs with high chromatic number

Abstract

We prove that for every nonnegative integer t and every graph G with maximum degree Δ(G) large enough with respect to t, if G has chromatic number χ(G)=Δ(G)−t then it contains a clique of size Δ(G)−2t2−6t−3. This generalizes (and provides an alternate proof for) a theorem of Cranston and Rabern, and of Mozhan, who proved the statement for the case t=0. Their result is the best known approximation to the famous Borodin-Kostochka Conjecture, which states that if χ(G)=Δ(G)≥9 then G should contain a Δ(G)-clique. Our result can also be viewed as a weak form of a statement conjectured by Reed, that quantifies more generally how large a clique a graph should contain if its chromatic number is close to its maximum degree.