$\chi$-binding function for a superclass of $2K_2$-free graphs

TL;DR

This paper investigates the chromatic number bounds for certain classes of graphs that exclude specific subgraphs, establishing optimal and linear $ ext{chi}$-binding functions for these classes, and identifying conditions for perfect graphs.

Contribution

It introduces the $inom{ ext{omega}+1}{2}$ $ ext{chi}$-binding function for a superclass of $2K_2$-free graphs and provides linear bounds and perfectness conditions for broader graph classes.

Findings

Connected $ ext{butterfly, hammer}$-free graphs with $ ext{omega} eq 2$ have $inom{ ext{omega}+1}{2}$ as a $ ext{chi}$-binding function.

Graphs free of $ ext{butterfly, hammer}$ and certain $H$-subgraphs admit linear $ ext{chi}$-binding functions.

Certain $ ext{butterfly, hammer, H}$-free graphs are perfect when $ ext{omega} extgreater= 2p$.

Abstract

The class of $2K_2$-free graphs has been well studied in various contexts in the past. In this paper, we study the chromatic number of $\{butterfly, hammer\}$-free graphs, a superclass of $2K_2$-free graphs and show that a connected $\{butterfly, hammer\}$-free graph $G$ with $\omega(G)\neq 2$ admits $\binom{\omega+1}{2}$ as a $\chi$-binding function which is also the best available $\chi$-binding function for its subclass of $2K_2$-free graphs. In addition, we show that if $H\in\{C_4+K_p, P_4+K_p\}$, then any $\{butterfly, hammer, H\}$-free graph $G$ with no components of clique size two admits a linear $\chi$-binding function. Furthermore, we also establish that any connected $\{butterfly, hammer, H\}$-free graph $G$ where $H\in \{(K_1\cup K_2)+K_p, 2K_1+K_p\}$, is perfect for $\omega(G)\geq 2p$.