On Bounding the Difference of the Maximum Degree and the Clique Number

For every k ∈ N0, we consider graphs in which for any induced subgraph, ∆ ≤ ω−1+k holds, where ∆ is the maximum degree and ω is the maximum clique number of the subgraph. We give a finite forbidden induced subgraph characterization for every k. As an application, we find some results on the chromatic number χ of a graph. B. Reed stated the conjecture that for every graph, χ ≤ ⌈ ∆+ω+1 2 ⌉ holds. Since this inequality is fulfilled by graphs in which ∆ ≤ ω + 2 holds, our results provide a hereditary graph class for which the conjecture holds.