AI 中文总结
研究不含\{P_6, 飞镖图, K_4\}的图的χ-界，通过证明此类图是6可着色的，改进了之前Hong和Xu的7的界，解决了最优性问题，还扩展了Karthick和Mishra关于不含\{P_6, 菱形图, K_4\}的图类的相关定理。
AI 中文摘要
菱形图是从K_4中去掉一条边得到的图，飞镖图是从菱形图中给一个3度顶点添加一条悬挂边得到的图。我们证明了每个不含\{P_6, 飞镖图, K_4\}的图都是6可着色的。这改进了Hong和Xu之前的7的界，并解决了他们关于该界最优性的开放问题。我们的结果还扩展了Karthick和Mishra的一个定理，他们证明了不含\{P_6, 菱形图, K_4\}的图类是6可着色的。
英文摘要
A \textit{diamond} is a graph obtained from \(K_4\) by removing an edge, and a \textit{dart} is a graph obtained from a diamond by adding a pendant edge to a vertex of degree 3. We prove that every $\{P_6, \text{dart}, K_4\}$-free graph is 6-colorable. This improves the previous bound of 7 due to Hong and Xu \cite{HongXu2025} and resolves their open question on the optimality of the bound. Our result also extends a theorem of Karthick and Mishra~\cite{KarthickMishra2018}, who proved 6-colorability for the class of \(\{P_6, \text{diamond}, K_4\}\)-free graphs.
Comments10 pages, comments are welcome
