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.

中文翻译：

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没有诱导五顶点路径或宝石的着色图

对于图$G$，令$\chi(G)$和$\omega(G)$分别表示$G$的色数和团数。我们给出了 ($P_5$,gem)-free 图的显式结构描述，并表明每个这样的图 $G$ 满足 $\chi(G)\le \lceil\frac{5\omega(G)}{4 }\rceil$。而且，这个界限是最好的。