Coloring of Some Crown-Free Graphs

Abstract

Let G and H be two vertex disjoint graphs. The union $G\cup H$ is the graph with $V(G\cup H)=V(G)\cup (H)$ and $E(G\cup H)=E(G)\cup E(H)$. The join $G+H$ is the graph with $V(G+H)=V(G)\cup V(H)$ and $E(G+H)=E(G)\cup E(H)\cup \{xy\;|\; x\in V(G), y\in V(H)$ $\}$. We use $P_k$ to denote a path on k vertices, use fork to denote the graph obtained from $K_{1,3}$ by subdividing an edge once, and use crown to denote the graph $K_1+K_{1,3}$. In this paper, we show that (i) $\chi (G)\le \frac{3}{2}(\omega ^2(G)-\omega (G))$ if G is (crown, $P_5$)-free, (ii) $\chi (G)\le \frac{1}{2}(\omega ^2(G)+\omega (G))$ if G is (crown, fork)-free, and (iii) $\chi (G)\le \frac{1}{2}\omega ^2(G)+\frac{3}{2}\omega (G)+1$ if G is (crown, $P_3\cup P_2$)-free.