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