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On the Erdös-Lovász Tihany Conjecture for Claw-Free Graphs

In 1968, Erdös and Lovász conjectured that for every graph $G$ and all integers $s,t\geq 2$ such that $s+t-1=χ(G) > ω(G)$, there exists a partition $(S,T)$ of the vertex set of $G$ such that $χ(G|S)\geq s$ and $χ(G|T)\geq t$. For general graphs, the only settled cases of the conjecture are when $s$ and $t$ are small. Recently, the conjecture was proved for a few special classes of graphs: graphs with stability number 2 \cite{quasi-line}, line graphs \cite{line} and quasi-line graphs \cite{quasi-line}. In this paper, we consider the conjecture for claw-free graphs and present some progress on it.