a1 Department of Mathematics, London School of Economics, Houghton Street, London WC2A 2AE, UK (e-mail: firstname.lastname@example.org)
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.
(Received July 18 2007)
(Revised April 02 2008)
(Online publication June 13 2008)