Monochromatic cycle covers in random graphs

TL;DR

This paper extends a classic monochromatic cycle cover result from complete graphs to random graphs, showing that with high probability, such covers exist with a bounded number of cycles under certain edge probability conditions.

Contribution

It introduces the study of monochromatic cycle covers in random graphs and establishes bounds on the number of cycles needed for coverage in this setting.

Findings

For p ≥ n^{-1/r + ε}, with high probability, G(n,p) admits monochromatic cycle covers with O(r^8 log r) cycles.

The bounds are nearly optimal; below p ≈ (log n / n)^{1/r}, the number of cycles needed can grow with n.

The results connect classical graph coloring theorems to the probabilistic setting of random graphs.

Abstract

A classic result of Erd\H{o}s, Gy\'arf\'as and Pyber states that for every coloring of the edges of $K_n$ with $r$ colors, there is a cover of its vertex set by at most $f(r) = O(r^2 \log r)$ vertex-disjoint monochromatic cycles. In particular, the minimum number of such covering cycles does not depend on the size of $K_n$ but only on the number of colors. We initiate the study of this phenomena in the case where $K_n$ is replaced by the random graph $\mathcal G(n,p)$. Given a fixed integer $r$ and $p =p(n) \ge n^{-1/r + \varepsilon}$, we show that with high probability the random graph $G \sim \mathcal G(n,p)$ has the property that for every $r$-coloring of the edges of $G$, there is a collection of $f'(r) = O(r^8 \log r)$ monochromatic cycles covering all the vertices of $G$. Our bound on $p$ is close to optimal in the following sense: if $p\ll (\log n/n)^{1/r}$, then with high probability there are colorings of $G\sim\mathcal G(n,p)$ such that the number of monochromatic cycles needed to cover all vertices of $G$ grows with $n$.