Some Cases of the Erd{o}s-Lov\'asz Tihany Conjecture for Claw-free Graphs

The Erd{o}s-Lov\'asz Tihany Conjecture states that any G with chromatic number χ(G) = s + t - 1 > ω(G), with s,t ≥ 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 ≤ s + 2, whenever G has a Kₛ, and for pairs $(s, t)$ if t ≤ 4 s - 3, whenever G contains a Kₛ and is claw-free. We also prove the Erd{o}s Lov\'asz Tihany Conjecture for the pair $(3, 10)$ for claw-free graphs.