Bounding χχ in terms of ωω and ΔΔ for some classes of graphs

Reed [B. Reed, ω,Δω,Δ and χχ, Journal of Graph Theory, 27 (1998) 177–212] conjectured that for any graph GG, χ(G)≤⌈Δ(G)+ω(G)+12⌉, where χ(G)χ(G), ω(G)ω(G), and Δ(G)Δ(G) respectively denote the chromatic number, the clique number and the maximum degree of GG. In this paper, we verify this conjecture for some special classes of graphs which are defined by families of forbidden induced subgraphs.