A rainbow version of Lehel's conjecture

Abstract

Lehel's conjecture states that every 2-edge-colouring of K_n admits a partition of its vertex set into two monochromatic cycles. It was proven for sufficiently large n by {\L}uczak, R\"odl, and Szemer\'edi in 1998, later improved by Allen in 2008, and fully resolved by Bessy and Thomass\'e in 2010. In this paper, we consider a rainbow analogue of Lehel's conjecture in the setting of properly edge-coloured complete graphs. We prove that, for sufficiently large n, every properly edge-coloured Kn admits a partition of its vertex set into two vertex-disjoint rainbow cycles