# Minimum degree conditions for monochromatic cycle partitioning

## TL;DR

This paper establishes the minimum degree threshold needed for an r-edge-coloured graph to be partitioned into a bounded number of monochromatic cycles, extending classical results and providing tight constructions.

## Contribution

It determines the minimum degree condition for monochromatic cycle partitioning in edge-coloured graphs, refining previous bounds and showing near-tightness.

## Key findings

- Minimum degree threshold at n/2 + c·r·log n for monochromatic cycle partitioning
- Partition into O(r^2) monochromatic cycles under the threshold
- Constructed examples demonstrating the tightness of the bounds

## Abstract

A classical result of Erd\H{o}s, Gy\'arf\'as and Pyber states that any $r$-edge-coloured complete graph has a partition into $O(r^2 \log r)$ monochromatic cycles. Here we determine the minimum degree threshold for this property. More precisely, we show that there exists a constant $c$ such that any $r$-edge-coloured graph on $n$ vertices with minimum degree at least $n/2 + c \cdot r \log n$ has a partition into $O(r^2)$ monochromatic cycles. We also provide constructions showing that the minimum degree condition and the number of cycles are essentially tight.