Structure and coloring of a family of ($P_7,C_5$)-free graphs

Abstract

Let $P_t$ and $C_t$ be a path and a cycle on $t$ vertices, respectively. In 2021, Choudum {\em et al.} [Disc. Math. 344 (2021) 112244] determined the structures of $(P_7,C_7,C_4$, diamond)-free and $(P_7,C_7,C_4$, gem)-frees, and gave correspondingly tight upper bounds to the chromatic numbers of these graphs. In this paper, we study the structure of $(P_7,C_5$, kite, paraglider)-free graphs, which is a superfamily of $(P_7,C_5$, diamond)-free graphs. We show that there is a unique connected imperfect $(P_7,C_5$, kite, paraglider)-free graph with $\delta (G)\ge \omega (G)+1$, which has no clique cutsets, no universal cliques, and no pair of vertices of which one's neighborhoods contains the other's. As a consequence, we show that $(P_7,C_5$, kite, paraglider)-free graphs are $\chi$-polydet with a binding function $\omega (G)+1$. Where a {\em diamond} (resp. {\em gem}) consists of a $P_3$ (resp. $P_4$) and a new vertex adjacent to all vertices of the $P_3$ (resp. $P_4$), a {\em kite} consists of a $P_4$ and a new vertex adjacent to consecutive three vertices of the $P_4$, and a {\em paraglider} consists of a $C_4$ and a new vertex adjacent to three vertices of the $C_4$.