# Chromatic bounds for some classes of $2K_2$-free graphs

## TL;DR

This paper investigates chromatic bounds for specific classes of $2K_2$-free graphs, establishing linear $ ext{chi}$-binding functions for certain subclasses and exploring $ ext{chi}$-boundedness of superclasses.

## Contribution

It identifies new classes of $2K_2$-free graphs that admit linear $ ext{chi}$-binding functions and extends $ ext{chi}$-boundedness results to some superclasses.

## Key findings

- Certain ($2K_2, H$)-free graph classes have linear $ ext{chi}$-binding functions.
- Some superclasses of $2K_2$-free graphs are $ ext{chi}$-bounded.
- The class of ($2K_2, 3K_1$)-free graphs does not admit a linear $ ext{chi}$-binding function.

## Abstract

A hereditary class $\mathcal{G}$ of graphs is $\chi$-bounded if there is a $\chi$-binding function, say $f$ such that $\chi(G) \leq f(\omega(G))$, for every $G \in \cal{G}$, where $\chi(G)$ ($\omega(G)$) denote the chromatic (clique) number of $G$. It is known that for every $2K_2$-free graph $G$, $\chi(G) \leq \binom{\omega(G)+1}{2}$, and the class of ($2K_2, 3K_1$)-free graphs does not admit a linear $\chi$-binding function. In this paper, we are interested in classes of $2K_2$-free graphs that admit a linear $\chi$-binding function. We show that the class of ($2K_2, H$)-free graphs, where $H\in \{K_1+P_4, K_1+C_4, \overline{P_2\cup P_3}, HVN, K_5-e, K_5\}$ admits a linear $\chi$-binding function. Also, we show that some superclasses of $2K_2$-free graphs are $\chi$-bounded.