无（fork，odd parachute）图中的三单纯顶点
Trisimplicial vertices in (fork, odd parachute)-free graphs
AI总结：
本文证明了非完美可分的(fork, odd parachute)-free图必含三单纯顶点，从而得出此类图的色数上界，并推广了现有结果及得出关于claw-free图的推论。
摘要：
图中的{\em奇洞}是一个长度至少为5的奇圈导出子图。{\em奇降落伞}是由奇洞$H$通过添加一条新边$uv$得到的图，使得对于每个$x\in V(H)$，$x$与$u$相邻但不与$v$相邻。如果对于图$G$的每个导出子图$H$，$V(H)$可以划分为$A$和$B$，使得$H[A]$是完美的且$ω(H[B])<ω(H)$，则图$G$是完美可分的。如果一个顶点的邻域是三个团的并集，则该顶点是{\em三单纯}的。本文证明了如果$G$是(fork, odd parachute)-free图，则$χ(G)\leq \binom{ω(G)+1}{2}$，证明方法是通过证明当$G$是非完美可分时，$G$包含一个三单纯顶点。这推广了Karthick, Kaufmann和Sivaraman [{\em Electron. J. Combin.} \textbf{29} (2022) \\#P3.19]，以及Wu和Xu [{\em Discrete Math.} \textbf{347} (2024) 114121]的一些结果。作为推论，每个非完美可分的claw-free图都包含一个三单纯顶点。
英文摘要：
An {\em odd hole} in a graph is an induced subgraph which is a cycle of odd length at least five. An {\em odd parachute} is a graph obtained from an odd hole $H$ by adding a new edge $uv$ such that $x$ is adjacent to $u$ but not to $v$ for each $x\in V(H)$. A graph $G$ is perfectly divisible if for each induced subgraph $H$ of $G$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $ω(H[B])<ω(H)$. A vertex of a graph is {\em trisimplicial} if its neighbourhood is the union of three cliques. In this paper, we prove that $χ(G)\leq \binom{ω(G)+1}{2}$ if $G$ is a (fork, odd parachute)-free graph by showing that $G$ contains a trisimplicial vertex when $G$ is nonperfectly divisible. This generalizes some results of Karthick, Kaufmann and Sivaraman [{\em Electron. J. Combin.} \textbf{29} (2022) \#P3.19], and Wu and Xu [{\em Discrete Math.} \textbf{347} (2024) 114121]. As a corollary, every nonperfectly divisible claw-free graph contains a trisimplicial vertex.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
