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

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 $ω(G)\neq 2$ admits $\binom{ω+1}{2}$ as a $χ$-binding function which is also the best available $χ$-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 $χ$-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 $ω(G)\geq 2p$.

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