A NOTE ON $\chi $-BINDING FUNCTIONS AND LINEAR FORESTS

Abstract Let G and H be two vertex disjoint graphs. The join $G+H$ is the graph with $V(G+H)=V(G)+V(H)$ and $E(G+H)=E(G)\cup E(H)\cup \{xy\;|\; x\in V(G), y\in V(H)\}$ . A (finite) linear forest is a graph consisting of (finite) vertex disjoint paths. We prove that for any finite linear forest F and any nonnull graph H , if $\{F, H\}$ -free graphs have a $\chi $ -binding function $f(\omega )$ , then $\{F, K_n+H\}$ -free graphs have a $\chi $ -binding function $kf(\omega )$ for some constant k .