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 afore-mentioned authors and by DeBiasio and Nelsen.