Balogh–Barát–Gerbner–Gyárfás–Sárközy cycle-partition conjecture

Let $G$ be a graph on $n$ vertices with a red-blue coloring of its edges. A partition into cycles is a collection of vertex-disjoint monochromatic cycles spanning $V(G)$.

Balogh–Barát–Gerbner–Gyárfás–Sárközy conjecture. If

$$\delta(G)>\frac{3n}{4},$$

then $G$ has a partition into a red cycle and a blue cycle.

This is the minimum-degree analogue of Lehel's conjecture for complete graphs. The paper proves the result under the asymptotically weaker condition $\delta(G)>(3/4+o(1))n$, while the exact threshold stated here remains open in the source.