Some Cases of the Erdos-Lov\'asz Tihany Conjecture for Claw-free Graphs

The Erdos-Lov\'asz Tihany Conjecture states that any G with chromatic number (G) = s + t - 1 > (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 Erdos Lov\'asz Tihany Conjecture for the pair (3, 10) for claw-free graphs.