Heterochromatic Matchings in Edge-Colored Graphs

Abstract

Let $G$ be an (edge-)colored graph. A heterochromatic matching of $G$ is a matching in which no two edges have the same color. For a vertex $v$, let $d^c(v)$ be the color degree of $v$. We show that if $d^c(v)\geq k$ for every vertex $v$ of $G$, then $G$ has a heterochromatic matching of size $\big\lceil{5k-3\over 12}\big\rceil$. For a colored bipartite graph with bipartition $(X,Y)$, we prove that if it satisfies a Hall-like condition, then it has a heterochromatic matching of cardinality $\big\lceil{|X|\over 2}\big\rceil$, and we show that this bound is best possible.