A hole in a graph is an induced cycle of length at least $4$. Let $sge2$ and $tge2$ be integers. A graph $G$ is $(s,t)$-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 ErdH{o}s-Lovasz Tihany Conjecture from 1968 states that every graph $G$ with $omega(G) < chi(G) = s + t - 1$ is $(s,t)$-splittable. This conjecture is hard, and few related results are known. However, it has been verified to be true for line graphs, quasi-line graphs, and graphs with independence number $2$. In this paper, we establish more evidence for the ErdH{o}s-Lovasz Tihany Conjecture by showing that every graph $G$ with $alpha(G)ge3$, $omega(G) < chi(G) = s + t - 1$, and no hole of length between $4$ and $2alpha(G)-1$ is $(s,t)$-splittable, where $alpha(G)$ denotes the independence number of a graph $G$.
تحميل البحث