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arXiv · 2204.00496

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

Peter Allen·Julia Böttcher·Richard Lang·Jozef Skokan·Maya Stein
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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.

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Peter Allen, Julia Böttcher, Richard Lang, Jozef Skokan, Maya Stein. 2022-04-01. Partitioning a 2-edge-coloured graph of minimum degree $2n/3 + o(n)$ into three monochromatic cycles. https://doi.org/10.1016/j.ejc.2023.103838

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