In this paper we show that every ( $P_6$, diamond)‐free graph $G$ satisfies $\chi \left(G\right)\le \omega \left(G\right)+3$, where $\chi \left(G\right)$ and $\omega \left(G\right)$ are the chromatic number and clique number of $G$, respectively. Our bound is attained by the complement of the famous 27‐vertex Schläfli 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 $\left(P_6,K_3\right)$‐free graphs.

中文翻译：

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（无P 6，菱形）图的最优χ界

在本文中，我们显示了每个（ $P_6$，钻石）-无图 $G$ 满足 $\chi （G）\le \omega （G）+3$， 在哪里 $\chi （G）$ 和 $\omega （G）$ 是色数和集团数 $G$， 分别。我们的界限是通过著名的27顶点Schläfli图的补码来实现的。我们的结果统一了关于线性存在的先前已知结果$\chi$几个图类的绑定函数。我们的证明是基于通过强完美图定理将不完美（$P_6$，不含钻石的图形），对这些图形结构的仔细分析以及依赖于众所周知的3色特征的计算机搜索 $（P_6，{ķ}_3）$免费图表。