The Erdős–Lovász Tihany conjecture for quasi-line graphs

Erdős and Lovász conjectured in 1968 that for every graph G with χ ( G ) > ω ( G ) and any two integers s , t ≥ 2 with s + t = χ ( G ) + 1 , there is a partition ( S , T ) of the vertex set V ( G ) such that χ ( G [ S ] ) ≥ s and χ ( 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.