Coloring graphs with no induced five-vertex path or gem

Abstract

For a graph $G$, let $\ddot{I}‡(G)$ and $\ddot{I}‰(G)$ 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 $\ddot{I}‡(G)\le \lceil \frac{5\ddot{I}‰(G)}{4}\rceil$. Moreover, this bound is best possible.

This paper is dedicated to the memory of Professor Frederic Maffray