Some Cases of the Erdős-Lovász Tihany Conjecture for Claw-free Graphs

The Erdős-Lovász Tihany Conjecture states that any \(G\) with chromatic number \(\chi(G) = s + t - 1 > \omega(G)\), with \(s,t \geq 2\) can be split into two vertex-disjoint subgraphs of chromatic number \(s, t\) respectively. We prove this conjecture for pairs \((s, t)\) if \(t \leq s + 2\), whenever \(G\) has a \(K_s\), and for pairs \((s, t)\) if \(t \leq 4 s - 3\), whenever \(G\) contains a \(K_s\) and is claw-free. We also prove the Erdős Lovász Tihany Conjecture for the pair \((3, 10)\) for claw-free graphs.