Homogeneous sets, clique-separators, critical graphs, and optimal $χ$-binding functions

Abstract

Given a set H of graphs, let f ⋆ H : N>0 → N>0 be the optimal χ-binding function of the class of H-free graphs, that is,In this paper, we combine the two decomposition methods by homogeneous sets and clique-separators in order to determine optimal χ-bindung functions for subclasses of P5-free graphs and of (C5, C7, . . .)-free graphs. In particular, we prove the following for each ω ≥ 1:We also characterise, for each of our considered graph classes, all graphs G with χ(G) > χ(G − u) for each u ∈ V (G). From these structural results, we can prove Reed's conjecture -relating chromatic number, clique number, and maximum degree of a graph -for (P5, banner)-free graphs and for (P5, dart)-free graphs.