Coloring hammer-free graphs with Δ − 1 colors

Abstract

The Borodin-Kostochka conjecture says that for a connected graph G, if Δ(G)≥9, then χ(G)≤max⁡{Δ(G)−1,ω(G)}. In this paper, we prove that the conjecture holds for hammer-free graphs, where a hammer is the graph obtained by identifying one vertex of a triangle and one end vertex of an induced path with three vertices.