Monochromatic Cycle Partitions of $2$-Coloured Graphs with Minimum Degree $3n/4$

Abstract

Balogh, Barát, Gerbner, Gyárfás, and Sárközy made 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 this conjecture for large $n$, improving approximate results by the aforementioned authors and by DeBiasio and Nelsen.