  Balogh, Bar\'at, Gerbner, Gy\'arf\'as, and S\'ark\"ozy proposed the following
conjecture. 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 be
partitioned into two vertex-disjoint cycles, one of each colour. We prove that
this conjecture holds for $n$ large enough, improving approximate results by
the aforementioned authors and by DeBiasio and Nelsen.
