Enhancing the Erdős-Lovász Tihany Conjecture for graphs with independence number two

Abstract

Let $s\ge 2$ and $t\ge 2$ be integers. A graph $G$ is $(s,t)$-\emph{splittable} if $V(G)$ can be partitioned into two sets $S$ and $T$ such that $\chi (G[S])\ge s$ and $\chi (G[T])\ge t$. The well-known Erdős-Lovász Tihany Conjecture from 1968 states that every graph $G$ whose chromatic number $\chi (G)=s+t-1$ is more than its clique number $\omega (G)$ is $(s,t)$-splittable. In this paper, we prove an enhanced version of the Erdős-Lovász Tihany Conjecture for graphs with independence number two. That is, for every graph $G$ with $\chi (G)=s+t-1>\omega (G)+1$ is $(s,t+1)$-splittable. There are examples showing that this result is best possible.