A Relaxed Version of the Erdős–Lovász Tihany Conjecture

The Erdős–Lovász Tihany conjecture asserts that every graph G with ω ( G ) < χ ( G ) = s + t − 1( s , t ≥ 2 ) contains two vertex disjoint subgraphs G 1 and G 2 such that χ ( G 1 ) ≥ s and χ ( G 2 ) ≥ t . Under the same assumption on G , we show that there are two vertex disjoint subgraphs G 1 and G 2 of G such that (a) χ ( G 1 ) ≥ s andcol ( G 2 ) ≥ t or (b)col ( G 1 ) ≥ s and χ ( G 2 ) ≥ t . Here, χ ( G ) is the chromatic number of G , ω ( G ) is the clique number of G , and col( G ) is the coloring number of G .