Enhancing the Erd\H{o}s-Lov\'asz Tihany Conjecture for graphs with independence number two

Let $s\ge2$ and $t\ge2$ be integers. A graph $G$ is $(s,t)$-\emph{splittable} if $V(G)$ can be partitioned into two sets $S$ and $T$ such that $\chi(G[S])\geq s$ and $\chi(G[T])\geq t$. The well-known Erd\H{o}s-Lov\'asz 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 Erd\H{o}s-Lov\'asz 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.