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Lehel conjectured in the 1970s that every red and blue edge-coloured completegraph can be partitioned into two monochromatic cycles. This was confirmed in2010 by Bessy and Thomass\'e. However, the host graph $G$ does not have to becomplete. 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 andS\'ark\"ozy. This degree condition is asymptotically tight.  Here we continue this line of research, by proving that for every red andblue 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 essentiallytight.

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