Ore- and P\'osa-type conditions for partitioning $2$-edge-coloured graphs into monochromatic cycles

TL;DR

This paper extends conditions under which large 2-edge-coloured graphs can be partitioned into monochromatic cycles, weakening degree requirements and providing approximate solutions for more general degree sum conditions.

Contribution

It introduces weaker degree sequence conditions for monochromatic cycle partitions and proves approximate results under Ore-type degree sum conditions.

Findings

Partition into two monochromatic cycles under a weaker degree sequence condition.

Almost partition into three monochromatic cycles with Ore-type degree sum condition.

Generalizes previous results by relaxing degree constraints.

Abstract

In 2019, Letzter confirmed a conjecture of Balogh, Bar\'at, Gerbner, Gy\'arf\'as and S\'ark\"ozy, proving that every large $2$-edge-coloured graph $G$ on $n$ vertices with minimum degree at least $3n/4$ can be partitioned into two monochromatic cycles of different colours. Here, we propose a weaker condition on the degree sequence of $G$ to also guarantee such a partition and prove an approximate version. This resembles a similar generalisation to an Ore-type condition achieved by Bar\'at and S\'ark\"ozy. Continuing work by Allen, B\"ottcher, Lang, Skokan and Stein, we also show that if $\operatorname{deg}(u) + \operatorname{deg}(v) \geq 4n/3 + o(n)$ holds for all non-adjacent vertices $u,v \in V(G)$, then all but $o(n)$ vertices can be partitioned into three monochromatic cycles.