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