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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.04496 |
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Table of Contents:
- 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.