  The Erd\H{o}s-Lov\'asz 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 Erd\H{o}s Lov\'asz Tihany Conjecture for the
pair $(3, 10)$ for claw-free graphs.
