  Let $K_n$ be the complete graph with $n$ vertices and $c_1, c_2, ..., c_r$ be
$r$ different colors. Suppose we randomly and uniformly color the edges of
$K_n$ in $c_1, c_2, ..., c_r$. Then we get a random graph, denoted by
$\mathcal{K}_n^r$. In the paper, we investigate the asymptotic properties of
several kinds of monochromatic and heterochromatic subgraphs in
$\mathcal{K}_n^r$. Accurate threshold functions in some cases are also
obtained.
