Non-perfect \((P_5, C_5, K_5-e)\)-Free Graphs are 5-Colorable

Let G be a graph. We use \(\chi (G)\) $\chi (G)$ and \(\omega (G)\) $\omega (G)$ to denote the chromatic number and clique number of G, respectively. A \(P_5\) $P_5$ is a path on 5 vertices, a \(C_5\) $C_5$ is a cycle on 5 vertices, and a \(K_5-e\) $K_5-e$ is obtained by removing one edge from \(K_5\) $K_5$ . Chudnovsky and Sivaraman showed that \(\chi (G)\le 2^{\omega (G)-1}\) $\chi (G)\le 2^{\omega (G)-1}$ if G is ( \(P_5, C_5)\) $P_5,C_5)$ -free. In this paper, we show that every non-perfect \((P_5, C_5, K_5-e)\) $(P_5,C_5,K_5-e)$ -free graph is 5-colorable.