  Given an edge-colored complete graph $K_n$ on $n$ vertices, a perfect
(respectively, near-perfect) matching $M$ in $K_n$ with an even (respectively,
odd) number of vertices is rainbow if all edges have distinct colors. In this
paper, we consider an edge coloring of $K_n$ by circular distance, and we
denote the resulting complete graph by $K^{\bullet}_n$. We show that when
$K^{\bullet}_n$ has an even number of vertices, it contains a rainbow perfect
matching if and only if $n=8k$ or $n=8k+2$, where $k$ is a nonnegative integer.
In the case of an odd number of vertices, Kirkman matching is known to be a
rainbow near-perfect matching in $K^{\bullet}_n$. However, real-world
applications sometimes require multiple rainbow near-perfect matchings. We
propose a method for using a recursive algorithm to generate multiple rainbow
near-perfect matchings in $K^{\bullet}_n$.
