Monochromatic cycle partitions of $r$-edge-coloured graphs with high minimum degree

TL;DR

This paper determines an almost optimal bound on the number of monochromatic cycles needed to cover all vertices in an r-edge-coloured graph with high minimum degree, resolving a question about monochromatic cycle partitions.

Contribution

It proves an upper bound on the number of monochromatic cycles needed, nearly matching the lower bounds, and disproves a previous conjecture about monochromatic tree covering.

Findings

Established an upper bound of O(r log r * ceil(r / log(1/δ))) for monochromatic cycle cover.

Constructed graphs showing the bound is tight up to a log r factor.

Disproved a conjecture of Bal and DeBiasio regarding monochromatic tree covering.

Abstract

A question posed independently by Letzter and Pokrovskiy asks: how many vertex-disjoint monochromatic cycles are needed to cover the vertex set of an $r$-edge-coloured graph, as a function of its minimum (uncoloured) degree? We resolve this problem up to a $(\log r)$-factor. Specifically, we prove that, for any $r \geq 2$ and $\delta \in (0,1/2)$, any $n$-vertex $r$-edge-coloured graph $G$ with $\delta(G) \geq (1- \delta)n$ can be covered with $\mathcal{O}(r \log r \cdot \lceil r/\log(1/\delta)\rceil)$ vertex-disjoint monochromatic cycles. We construct graphs that show this is tight up to the $(\log r)$-factor for all values of $r$ and $\delta$, and along the way disprove a conjecture of Bal and DeBiasio about monochromatic tree covering.