Coloring graphs with no induced five-vertex path or gem

TL;DR

This paper provides a structural characterization of ($P_5$,gem)-free graphs and establishes a tight upper bound on their chromatic number relative to their clique number, advancing understanding in graph coloring.

Contribution

It offers an explicit structural description of ($P_5$,gem)-free graphs and proves a tight upper bound on their chromatic number based on clique number.

Findings

($P_5$,gem)-free graphs have a specific structure.

The chromatic number satisfies $oxed{ ext{chi}(G) \\le \\lceil rac{5 \\omega(G)}{4} ceil}$.

The bound is proven to be optimal.

Abstract

For a graph $G$, let $\chi(G)$ and $\omega(G)$ respectively denote the chromatic number and clique number of $G$. We give an explicit structural description of ($P_5$,gem)-free graphs, and show that every such graph $G$ satisfies $\chi(G)\le \lceil\frac{5\omega(G)}{4}\rceil$. Moreover, this bound is best possible.