On bounding the difference between the maximum degree and the chromatic number by a constant

Abstract

We provide a finite forbidden induced subgraph characterization for the graph class Υk, for all k∈N0, which is defined as follows. A graph is in Υk if for any induced subgraph, Δ≤χ−1+k holds, where Δ is the maximum degree and χ is the chromatic number of the subgraph.We compare these results with those given in Schaudt and Weil (2015), where we studied the graph class Ωk, for k∈N0, whose graphs are such that for any induced subgraph, Δ≤ω−1+k holds, where ω denotes the clique number of a graph. In particular, we give a characterization in terms of Ωk and Υk of those graphs where the neighborhood of every vertex is perfect.