Structure and coloring of (P7, C5, diamond)-free graphs

Abstract

Let $P_t$ and $C_t$ denote a path and a cycle on $t$ vertices, respectively. A diamond consists of two triangles sharing exactly one edge, a paw is a graph obtained from a triangle by adding a pendant edge. Let $H$ be a diamond or a paw. In this paper, we determine the structure of imperfect ($P_7,C_5,H$)-free graphs. As a consequence, we show that each ($P_7,C_5,H$)-free graph $G$ can be polynomially colored with $max\{3,\omega \left(G\right)\}$ colors. We also show that each ($P_7,C_5$, bull)-free graph is perfectly divisible, where a bull is a graph consisting of a triangle with two disjoint pendant edges. As a consequence, $\chi \left(G\right)\le \left(\frac{\omega \left(G\right)+1}{2}\right)$ if $G$ is ($P_7,C_5$, bull)-free.