On minimal nonperfectly divisible fork-free graphs

TL;DR

This paper addresses whether fork-free graphs are perfectly divisible by establishing that minimal nonperfectly divisible fork-free graphs must be claw-free, effectively reducing the problem to claw-free structure. It then shows that (fork, $P_7$)-free and (fork, $P_6 \cup K_1$)-free graphs are perfectly divisible by analyzing odd holes within the neighborhood $M(v)$ and deriving a perfect division under these forbidden configurations. Consequently, these classes satisfy the chromatic bound $\\chi(G) \\leq \\binom{\\omega(G)+1}{2}$. The results advance the fork-free conjecture by isolating the minimal nonperfectly divisible case to claw-free graphs and providing strong bounds for two important forbidden-subgraph families.

Abstract

A fork is a graph obtained from $K_{1,3}$ (usually called claw) by subdividing an edge once. A graph is perfectly divisible if for each of its induced subgraph $H$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $ω(H[B]) < ω(H)$. In this paper, we prove that the perfect divisibility of fork-free graphs is equivalent to that of claw-free graphs. We also prove that, for $F\in \{P_7, P_6\cup K_1\}$, each (fork, $F$)-free graph $G$ is perfectly divisible and hence $χ(G)\leq \binom{ω(G)+1}{2}$.

Figures (2)

• Figure 1: Illustration of fork and some forbidden configurations.
• Figure 2: A simple illustration of the sets $V(C), U, U', Z, Z'$ and $M(C)$ etc.

Theorems & Definitions (30)

• Conjecture 1.1
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5