On the relation among the maximum degree, the chromatic number and the clique number

Abstract

We study the relation among the maximum degree $\Delta(G)$, the chromatic number $\chi(G)$ and the clique number $\omega(G)$ of a graph $G$ by proving results related to the Borodin-Kostochka Conjecture and Reed's Conjecture, which are two long-standing conjectures on this subject. The Borodin-Kostochka Conjecture states that a graph $G$ with $\chi(G)=\Delta(G)\ge9$ has a clique of size $\Delta(G)$. Reed's Conjecture states that any graph $G$ satisfies $\chi(G)\le \lceil\frac{1}{2}(\Delta(G)+\omega(G)+1)\rceil$.

By combining and extending the approaches taken by Cranston and Rabern and by Haxell and MacDonald, we use the technique of Mozhan partitions to prove a generalized Borodin-Kostochka-type result as follows. Given a nonnegative integer $t$, for every graph $G$ with $\Delta(G)\ge 4t^2+11t+7$ and $\chi(G)=\Delta(G)-t$, the graph $G$ contains a clique of size $\Delta(G)-2t^2-7t-4$.

We further prove that both conjectures hold for graphs with at least one critical vertex that is not in an odd hole, by following similar proofs as by Aravind, Karthick and Subramanian and by Chen, Lan, Lin and Zhou. This result then motivates us to study Reed's Conjecture in the context of graphs in which all vertices are in some odd hole. By investigating blow-ups and powers of cycles, we enumerate several classes of graphs for which Reed's Conjecture holds with equality. They generalize many of the known tight examples. We also introduce a class of graphs constructed from blow-ups and powers of paths, which gives rise to a new family of irregular tight examples.