Trisimplicial vertices in (fork, odd parachute)-free graphs

TL;DR

This paper addresses the chromatic-bounding problem for the class of $(\text{fork}, \text{odd parachute})$-free graphs. It develops a structural framework showing that any nonperfectly divisible graph in this class contains a trisimplicial vertex, established via a detailed analysis around minimal odd balloons and forbidden subgraph configurations. The main result implies the bound $\\chi(G) \\le \\binom{\\omega(G)+1}{2}$, generalizing prior work and yielding corollaries such as the existence of trisimplicial vertices in nonperfectly divisible claw-free graphs. These findings advance the understanding of $\\chi$-boundedness in claw-free, fork-free, and related graph families and connect to broader questions in perfect divisibility and graph coloring. The work provides a concrete structural mechanism that links local neighborhood decompositions to global coloring bounds, with potential implications for related hereditary graph classes.

Abstract

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.

Figures (3)

• Figure 1: Illustrations of the configurations.
• Figure 2: 5-balloon and 5-parachute.
• Figure 3: Illustrations of $Q'=F_1\cup F_2$.

Theorems & Definitions (13)

• Theorem 1.1: Chudnovsky-Seymour MC2010
• Conjecture 1.2: Sivaraman TKJK2022
• Theorem 1.3: Wu-Xu DW2023
• Theorem 1.4: Wu-Xu DW2023
• Theorem 1.5
• Corollary 1.6
• Corollary 1.7
• Lemma 2.1: Wu-Xu DW2023
• proof : Proof of Theorem \ref{['forkgem+tri']}
• proof : Proof of Corollary \ref{['MTHM']}