Bounding in terms of and for some classes of graphs

Reed [B. Reed, ω , Δ and χ , Journal of Graph Theory, 27 (1998) 177–212] conjectured that for any graph G , χ ( G ) ≤ ⌈ Δ ( G ) + ω ( G ) + 1 2 ⌉ , where χ ( G ) , ω ( G ) , and Δ ( G ) respectively denote the chromatic number, the clique number and the maximum degree of G . In this paper, we verify this conjecture for some special classes of graphs which are defined by families of forbidden induced subgraphs.