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

For every k ∈ N0, we consider graphs G where for every induced subgraph H of G, ∆(H) ≤ χ(H) − 1 + k holds, where ∆(H) is the maximum degree and χ(H) is the chromatic number of the subgraph H . Let us call this family of graphs Υk. We give a finite forbidden induced subgraph characterization of Υk for every k. We compare these results with those given in On bounding the difference between the maximum degree and the clique number, Graphs and Combinatorics 31(5): 1689-1702 (2015), where we studied the graphs in which for any induced subgraphH , ∆(H) ≤ ω(H)−1+k holds, where ω(H) denotes the clique number of a graph. In particular, we introduce the class of neighborhood perfect graphs, that is, those graphs where the neighborhood of every vertex is perfect. We find a nice characterization of this graph class in terms of Ωk and Υk: We prove that a graph G is a neighborhood perfect graph if and only if for every induced subgraph H of G, H ∈ Υk if and only if H ∈ Ωk for all k ∈ N0.