An Optimal $χ$-Bound for ($P_6$, diamond)-Free Graphs

Given two graphs $H1$ and $H2$, a graph $G$ is $(H1,H2)$-free if it contains no induced subgraph isomorphic to $H1$ or $H2$. Let $Pt$ be the path on $t$ vertices and $Kt$ be the complete graph on $t$ vertices. The diamond is the graph obtained from $K4$ by removing an edge. In this paper we show that every ($P6$, diamond)-free graph $G$ satisfies $\chi(G)\le \omega(G)+3$, where $\chi(G)$ and $\omega(G)$ are the chromatic number and clique number of $G$, respectively. Our bound is attained by the complement of the famous 27-vertex Schl\"afli graph. Our result unifies previously known results on the existence of linear $\chi$-binding functions for several graph classes. Our proof is based on a reduction via the Strong Perfect Graph Theorem to imperfect ($P6$, 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 $(P6,K_3)$-free graphs.