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$\chi $-BINDING FUNCTIONS AND LINEAR FORESTS
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.
