Perfect divisibility of (fork, antifork$\cup K_1$)-free graphs

TL;DR

The paper proves that the class of $(fork, antifork \cup K_1)$-free graphs are perfectly divisible, and moreover satisfies a $\chi$-bound $\chi(G) \le \binom{\omega(G)+1}{2}$. The authors deploy a contradiction-based argument via a minimal counterexample, leveraging claw-freeness and a detailed hole-based structural decomposition around an odd hole to extend a perfect division from a subgraph $G[M(C_0)]$ to the entire graph. This establishes a strong structural result for fork-free related graph classes and extends the landscape of $\chi$-boundedness for hereditary graph families. The findings strengthen prior work on fork-free graphs and yield improved bounds in the $(fork, co-cricket)$-free setting as a corollary.

Abstract

A {\em fork} is a graph obtained from $K_{1,3}$ (usually called {\em claw}) by subdividing an edge once, an {\em antifork} is the complement graph of a fork, and a {\em co-cricket} is a union of $K_1$ and $K_4-e$. 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 $ω(H[B]) < ω(H)$. Karthick {\em et al.} [Electron. J. Comb. 28 (2021), P2.20.] conjectured that fork-free graphs are perfectly divisible, and they proved that each (fork, co-cricket)-free graph is either claw-free or perfectly divisible. In this paper, we show that every (fork, {\em antifork}$\cup K_1$)-free graph is perfectly divisible. This improves some results of Karthick {\em et al.}.

Figures (1)

• Figure 1: Illustration of fork and some forbidden configurations.

Theorems & Definitions (12)

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