Monochromatic cycle partitions of \(2\)-coloured graphs with minimum degree \(3n/4\) (Q668072)

Summary: \textit{J. Balogh} et al. [Combinatorica 34, No. 5, 507--526 (2014; Zbl 1340.05069)] 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 Balogh et al. [loc. cit.] and by \textit{L. DeBiasio} and \textit{L. L. Nelsen} [J. Comb. Theory, Ser. B 122, 634--667 (2017; Zbl 1350.05135)].