The chromatic number of $\{P_6,\text{banner, diamond}\}$-free graphs

Abstract

Let $P_6$ denote a 6-vertex path, a \textit{banner} be a 4-cycle with a pendant vertex, and a \textit{diamond} be $K_4$ minus an edge. We demonstrate that for any graph $G$ in the class of $\{P_6,\text{banner},\text{diamond}\}$-free graphs, the chromatic number is always bounded above by the maximum of 3 and the clique number. This strictly strengthens a result of Lan, Zhou and Liu [20] by replacing their $C_4$-free condition with the weaker banner-free requirement.