Structure and coloring of some ($P_7,C_4$)-free graphs

Abstract

Let $G$ be a graph. We use $P_t$ and $C_t$ to denote a path and a cycle on $t$ vertices, respectively. A {\em diamond} is a graph obtained from two triangles that share exactly one edge. A {\em kite} is a graph consists of a diamond and another vertex adjacent to a vertex of degree 2 of the diamond. A {\em gem} is a graph that consists of a $P_4$ plus a vertex adjacent to all vertices of the $P_4$. In this paper, we prove some structural properties to $(P_7, C_4,$ diamond)-free graphs, $(P_7, C_4,$ kite)-free graphs and $(P_7, C_4,$ gem)-free graphs. As their corollaries, we show that (\romannumeral 1) $\chi (G)\leq \max\{3,\omega(G)\}$ if $G$ is $(P_7, C_4,$ diamond)-free, (\romannumeral 2) $\chi(G)\leq \omega(G)+1$ if $G$ is $(P_7, C_4,$ kite)-free and (\romannumeral 3) $\chi(G)\leq 2\omega(G)-1$ if $G$ is $(P_7, C_4,$ gem)-free. These conclusions generalize some results of Choudum {\em et al} and Lan {\em et al}.