AI 中文总结
研究不含\((P_6,C_4)\)的图的着色问题，核心方法是在\(\Delta(G)\geq9\)且\(\omega(G)<\Delta(G)\)条件下，证明此类图是\((\Delta(G)-1)\)可着色的，为图的着色研究提供了新结论。
AI 中文摘要
对于图\(G\)，\(\Delta(G)\)、\(\omega(G)\)和\(\chi(G)\)分别表示其最大度、团数和色数。\(P_n\)和\(C_n\)分别表示\(n\)个顶点的无弦路径和无弦圈。本文证明了每个不含\((P_6,C_4)\)且\(\Delta(G)\geq9\)、\(\omega(G)<\Delta(G)\)的图\(G\)是\((\Delta(G)-1)\)可着色的。
英文摘要
For a graph $G$, let $Δ(G)$, $ω(G)$, and $χ(G)$ denote the maximum degree, clique number, and chromatic number of $G$, respectively. Let $P_n$ and $C_n$ denote the chordless path and chordless cycle on $n$ vertices, respectively. In this paper, we prove that every $(P_6,C_4)$-free graph $G$ with $Δ(G)\ge 9$ and $ω(G)<Δ(G)$ is $(Δ(G)-1)$-colorable.
