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arXiv · 2202.06388

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

Patrick Arras
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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.

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Patrick Arras. 2022-02-13. Ore- and P\'osa-type conditions for partitioning $2$-edge-coloured graphs into monochromatic cycles. https://doi.org/10.37236/11052

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