Partitioning a 2-edge-coloured graph of minimum degree $2n/3 + o(n)$ into three monochromatic cycles

TL;DR

This paper proves that in large graphs with minimum degree slightly above two-thirds of the number of vertices, the vertices can be partitioned into three monochromatic cycles, extending previous results on monochromatic cycle partitions.

Contribution

It establishes a near-tight minimum degree condition for partitioning a 2-edge-coloured graph into three monochromatic cycles, advancing the understanding of monochromatic cycle decompositions.

Findings

Partition into three monochromatic cycles for minimum degree ≥ 2n/3 + o(n)

Extends previous results from complete and higher degree graphs

Nearly tight bound confirmed for the problem

Abstract

Lehel conjectured in the 1970s that every red and blue edge-coloured complete graph can be partitioned into two monochromatic cycles. This was confirmed in 2010 by Bessy and Thomass\'e. However, the host graph $G$ does not have to be complete. It it suffices to require that $G$ has minimum degree at least $3n/4$, where $n$ is the order of $G$, as was shown recently by Letzter, confirming a conjecture of Balogh, Bar\'{a}t, Gerbner, Gy\'arf\'as and S\'ark\"ozy. This degree condition is asymptotically tight. Here we continue this line of research, by proving that for every red and blue edge-colouring of an $n$-vertex graph of minimum degree at least $2n/3 + o(n)$, there is a partition of the vertex set into three monochromatic cycles. This approximately verifies a conjecture of Pokrovskiy and is essentially tight.