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

TL;DR

This paper establishes a tight upper bound on the chromatic number for ($P_6$, diamond)-free graphs, unifying previous results and using structural analysis combined with computational methods.

Contribution

It proves that ($P_6$, diamond)-free graphs satisfy $ ext{chi}(G) extless= ext{omega}(G)+3$, providing the optimal bound and connecting to the Schl"afli graph.

Findings

The bound $ ext{chi}(G) extless= ext{omega}(G)+3$ is tight.

The result generalizes and unifies earlier bounds for related graph classes.

The proof combines structural graph theory, the Strong Perfect Graph Theorem, and computer-aided analysis.

Abstract

Given two graphs $H_1$ and $H_2$, a graph $G$ is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ or $H_2$. Let $P_t$ be the path on $t$ vertices and $K_t$ be the complete graph on $t$ vertices. The diamond is the graph obtained from $K_4$ by removing an edge. In this paper we show that every ($P_6$, 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 ($P_6$, 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.