Monochromatic cycle partitions of graphs with large minimum degree

Balogh, Barát, Gerbner, Gyárfás, and Sárközy proposed a significant strengthening of Lehel's conjecture where Kn is replaced by any graph G with δ(G)>3n/4; if true, this minimum degree condition is essentially best possible. We prove that their conjecture holds when δ(G)>(3/4+o(1))n. Our proof uses Szemerédi's regularity lemma along with the absorbing method of Rödl, Ruciński, and Szemerédi by first showing that the graph can be covered with monochromatic subgraphs having certain robust expansion properties.