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Balogh, Bar\'at, Gerbner, Gy\'arf\'as, and S\'ark\"ozy proposed the followingconjecture. Let $G$ be a graph on $n$ vertices with minimum degree at least$3n/4$. Then for every $2$-edge-colouring of $G$, the vertex set $V(G)$ may bepartitioned into two vertex-disjoint cycles, one of each colour. We prove thatthis conjecture holds for $n$ large enough, improving approximate results bythe aforementioned authors and by DeBiasio and Nelsen.

Monochromatic cycle partitions of $2$-coloured graphs with minimum degree $3n/4$