Monochromatic and heterochromatic subgraph problems in a randomly colored graph

Abstract

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.

11 pages