On Finding Large Cliques when the Chromatic Number is close to the Maximum Degree

Abstract

We prove that every graph G with chromatic number χ(G) = ∆(G) − 1 and ∆(G) ≥ 66 contains a clique of size ∆(G) − 17. Our proof closely parallels a proof from Cranston and Rabern, who showed that graphs with χ = ∆ and ∆ ≥ 13 contain a clique of size ∆ − 3. Their result is the best currently known for general ∆ towards the Borodin-Kostochka conjecture, which posits that graphs with χ = ∆ and ∆ ≥ 9 contain a clique of size ∆. We also outline some related progress which has been made towards the conjecture.