Monochromatic and heterochromatic subgraph problems in a randomly colored graph

TL;DR

This paper studies the asymptotic behavior of monochromatic and heterochromatic subgraphs in a randomly edge-colored complete graph, providing threshold functions for their appearance.

Contribution

It introduces new results on the thresholds and properties of subgraphs in randomly colored complete graphs, advancing understanding of their probabilistic structure.

Findings

Identifies threshold functions for the emergence of certain subgraphs.

Analyzes asymptotic properties of monochromatic and heterochromatic subgraphs.

Provides precise probabilistic bounds for subgraph appearances.

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.