An optimal χ‐bound for (P6, diamond)‐free graphs

Abstract

In this paper we show that every ( P 6, diamond)‐free graph G satisfies χ ( G ) ≤ ω ( G ) + 3, where χ ( G ) and ω ( G ) are the chromatic number and clique number of G, respectively. Our bound is attained by the complement of the famous 27‐vertex Schläfli graph. Our result unifies previously known results on the existence of linear χ‐binding functions for several graph classes. Our proof is based on a reduction via the Strong Perfect Graph Theorem to imperfect ( P 6, diamond)‐free graphs, a careful analysis of the structure of those graphs, and a computer search that relies on a well‐known characterization of 3‐colourable ( P 6 , K 3 )‐free graphs.

Colouring graphs with no induced six-vertex path or diamond

“…Our results are an improvement of the result of Cameron, Huang and Merkel [4], and answer an open question from [4]. We believe that our proof technique for polynomial time solvability may also be useful for other graph families (see Section 6).…”

“…This theorem is an improvement of the result of Cameron, Huang and Merkel [4] that χ(G) ≤ ω(G) + 3 for a (P 6 , diamond)-free graph G. To prove Theorem 5, we use Theorem 6 below. Theorem 6.…”

“…Let G = (V, E) be an imperfect (P 6 , diamond)-free graph. We follow the notation of [4] and partition V into the following subsets:…”

“…It is proved in [4] that if K has no neighbours in C, then K is 3-colourable. (See Claim 1 in [4]. The conditions that G is connected and has no clique cutsets or comparable vertices are necessary.)…”

“…In the same paper, they proved that every (P 6 , diamond, K 4 )-free graph is 6-colourable. Finally, Cameron, Huang, and Merkel [4] improved the χ-binding function of (P 6 , diamond)-free graphs to ω(G) + 3.…”