Chromatic number of P5P5-free graphs: Reed’s conjecture

In this paper we study the chromatic number of P5P5-free graphs. In 1998, Reed proposed the following Conjecture: Every graph GG with chromatic number χ(G)χ(G), clique number ω(G)ω(G) and maximum degree Δ(G)Δ(G) satisfies χ(G)≤⌈ω(G)+Δ(G)+12⌉. Reed’s conjecture is still open in general.Our main result is that Reed’s conjecture holds for the class of P5P5-free graphs in the asymptotic sense. We will also show that Reed’s conjecture is true for many special P5P5-free graphs. Moreover, we will present polynomial χχ-binding functions for these subclasses.