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

TL;DR

This paper proves that graphs free of both forks and antiforks (plus isolated vertices) are perfectly divisible, advancing the understanding of graph structure and coloring properties in this class.

Contribution

It establishes that (fork, antifork∪K_1)-free graphs are perfectly divisible, improving previous results on related graph classes.

Findings

Proves (fork, antifork∪K_1)-free graphs are perfectly divisible.

Extends the class of graphs known to be perfectly divisible.

Improves upon prior results by Karthick et al.

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.}.