The optimal $\chi$-bound for $\{P_6,\text{dart},K_4\}$-free graphs

A $\mathit{\text{diamond}}$ is a graph obtained from $K_4$ by removing an edge, and a $\mathit{\text{dart}}$ is a graph obtained from a diamond by adding a pendant edge to a vertex of degree 3. We prove that every $\{P_6,\text{dart},K_4\}$-free graph is 6-colorable. This improves the previous bound of 7 due to Hong and Xu \cite{HongXu2025} and resolves their open question on the optimality of the bound. Our result also extends a theorem of Karthick and Mishra~\cite{KarthickMishra2018}, who proved 6-colorability for the class of $\{P_6,\text{diamond},K_4\}$-free graphs.