Perfect divisibility and coloring of some fork-free graphs

TL;DR

This paper proves that certain fork-free graphs are perfectly divisible and establishes upper bounds on their chromatic number based on clique number, extending previous results in graph theory.

Contribution

It demonstrates perfect divisibility for (odd balloon, fork)-free graphs and provides chromatic bounds for (fork, gem)-free and (fork, butterfly)-free graphs.

Findings

(odd balloon, fork)-free graphs are perfectly divisible

Chromatic number bound: χ(G) ≤ (ω(G)+1 choose 2) for specific graph classes

Generalizes previous results by Karthick et al.

Abstract

A $hole$ is an induced cycle of length at least four, and an odd hole is a hole of odd length. A {\em fork} is a graph obtained from $K_{1,3}$ by subdividing an edge once. An {\em odd balloon} is a graph obtained from an odd hole by identifying respectively two consecutive vertices with two leaves of $K_{1, 3}$. A {\em gem} is a graph that consists of a $P_4$ plus a vertex adjacent to all vertices of the $P_4$. A {\em butterfly} is a graph obtained from two traingles by sharing exactly one vertex. 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 $\omega(H[B])<\omega(H)$. In this paper, we show that (odd balloon, fork)-free graphs are perfectly divisible (this generalizes some results of Karthick {\em et al}). As an application, we show that $\chi(G)\le\binom{\omega(G)+1}{2}$ if $G$ is (fork, gem)-free or (fork, butterfly)-free.