  For a given graph $H$ and $n\geq 1$, let $f(n,H)$ denote the maximum number
$c$ for which there is a way to color the edges of the complete graph $K_n$
with $c$ colors such that every subgraph $H$ of $K_n$ has at least two edges of
the same color. Equivalently, any edge-coloring of $K_n$ with at least
$rb(n,H)=f(n,H)+1$ colors contains a rainbow copy of $H$, where a rainbow
subgraph of an edge-colored graph is such that no two edges of it have the same
color. The number $rb(n,H)$ is called the {\it rainbow number of $H$}.
Erd\H{o}s, Simonovits and S\'{o}s showed that $rb(n,K_3)=n$. In 2004,
Schiermeyer used some counting technique and determined the rainbow numbers
$rb(n,kK_2)$ for $k\geq 2$ and $n\geq 3k+3$. It is easy to see that $n$ must be
at least $2k$. So, for $2k \leq n<3k+3$, the rainbow numbers remain not
determined. In this paper we will use the Gallai-Edmonds structure theorem for
matchings to determine the exact values for rainbow numbers $rb(n,kK_2)$ for
all $k\geq 2$ and $n\geq 2k$, giving a complete solution for the rainbow
numbers of matchings.

Combinatorics