Rainbow Matchings in Properly Edge Colored Graphs

Abstract

Let $G$ be a properly edge colored graph. A rainbow matching of $G$ is a matching in which no two edges have the same color. Let $\delta$ denote the minimum degree of $G$. We show that if $|V(G)|\geq \frac{8\delta}{5}$, then $G$ has a rainbow matching of size at least $\lfloor\frac {3 \delta }{5}\rfloor$. We also prove that if $G$ is a properly colored triangle-free graph, then $G$ has a rainbow matching of size at least $\lfloor\frac {2 \delta }{3}\rfloor$.