The Erdős–Lovász Tihany conjecture for quasi-line graphs

Erdős and Lovász conjectured in 1968 that for every graph GG with χ(G)>ω(G)χ(G)>ω(G) and any two integers s,t≥2s,t≥2 with s+t=χ(G)+1s+t=χ(G)+1, there is a partition (S,T)(S,T) of the vertex set V(G)V(G) such that χ(G[S])≥sχ(G[S])≥s and χ(G[T])≥tχ(G[T])≥t. Except for a few cases, this conjecture is still unsolved. In this note we prove the conjecture for quasi-line graphs and for graphs with independence number 2.