Coloring graph classes with no induced fork via perfect divisibility

TL;DR

This paper investigates the structure of fork-free graphs, establishing a quadratic bound on their chromatic number in relation to their clique number within certain classes, and explores perfect divisibility properties.

Contribution

It introduces a new quadratic bound for the chromatic number of (fork, F)-free graphs using perfect divisibility, advancing understanding of coloring in these graph classes.

Findings

Every (fork, F)-free graph G satisfies χ(G) ≤ ω(G)^2.

The class of (fork, F)-free graphs does not admit a linear χ-binding function.

The study links perfect divisibility with chromatic bounds in specific graph classes.

Abstract

For a graph $G$, $\chi(G)$ will denote its chromatic number, and $\omega(G)$ its clique number. A graph $G$ is said to be perfectly divisible if for all induced subgraphs $H$ of $G$, $V(H)$ can be partitioned into two sets $A$, $B$ such that $H[A]$ is perfect and $\omega(H[B]) < \omega(H)$. An integer-valued function $f$ is called a $\chi$-binding function for a hereditary class of graphs $\cal C$ if $\chi(G) \leq f(\omega(G))$ for every graph $G\in \cal C$. The fork is the graph obtained from the complete bipartite graph $K_{1,3}$ by subdividing an edge once. The problem of finding a polynomial $\chi$-binding function for the class of fork-free graphs is open. In this paper, we study the structure of some classes of fork-free graphs; in particular, we study the class of (fork,$F$)-free graphs $\cal G$ in the context of perfect divisibility, where $F$ is a graph on five vertices with a stable set of size three, and show that every $G\in \cal G$ satisfies $\chi(G)\leq \omega(G)^2$. We also note that the class $\cal G$ does not admit a linear $\chi$-binding function.