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arXiv · 2505.04429

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

Ran Chen·Baogang Xu·Miaoxia Zhuang
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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 $\omega(H[B]) < \omega(H)$. 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.}.

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Ran Chen, Baogang Xu, Miaoxia Zhuang. 2025-05-07. Perfect divisibility of (fork, antifork$\cup K_1$)-free graphs. https://arxiv.org/abs/2505.04429

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