Covering two-edge-coloured complete graphs with two disjoint monochromatic cycles

In 1998 Łuczak Rödl and Szemerédi proved, by means of the Regularity Lemma, that there exists n0 such that, for any n ≥ n0 and two-edge-colouring of Kn, there exists a pair of vertex-disjoint monochromatic cycles of opposite colours covering the vertices of Kn. In this paper we make use of an alternative method of finding useful structure in a graph, leading to a proof of the same result with a much smaller value of n0. The proof gives a polynomial-time algorithm for finding the two cycles.