Chromatic bounds for the subclasses of $pK_2$-free graphs

TL;DR

This paper advances the understanding of chromatic bounds in specific subclasses of $pK_2$-free graphs by providing improved and new $ ext{chi}$-binding functions, including linear and polynomial bounds, for various graph classes.

Contribution

It introduces new and improved $ ext{chi}$-binding functions for subclasses of $pK_2$-free graphs, including linear and polynomial bounds, with alternative proofs for existing classes.

Findings

Improved $ ext{chi}$-binding function for $ ext{ extless}2K_2, K_1 + C_4 ext{ extgreater}$-free graphs.

Linear $ ext{chi}$-binding function for $ extless}2K_2, K_2+P_4 extgreater}$-free graphs.

Polynomial $ ext{chi}$-binding functions for $ extless}pK_2, H extgreater}$-free graphs with $H$ in a specified set.

Abstract

The concept of $\chi$-binding functions for classes of free graphs has been extensively studied in the past. In this paper, we improve the existing $\chi$-binding function for $\{2K_2, K_1 + C_4\}$-free graphs. Also, we find a linear $\chi$-binding function for $\{2K_2, K_2+P_4\}$-free graphs. In addition, we give alternative proofs for the $\chi$-binding function of $\{2K_2,gem\}$-free graphs, $\{2K_2,HVN\}$-free graphs and $\{2K_2, K_5-e\}$-free graphs. Finally, for $p\geq3$, we find polynomial $\chi$-binding functions for $\{pK_2, H\}$-free graphs where $H\in \{gem, diamond, K_2+P_4, HVN, K_5-e, butterfly, gem^+, dart, K_1 + C_4, C_4, \overline{P_5}\}$.