A rainbow version of Lehel's conjecture
Pedro Araújo, Xiao-Chuan Liu, Taísa Martins, Walner Mendonça, Luiz Moreira, Vinícius Fernandes dos Santos, Zhifei Yan
Abstract
Lehel's conjecture states that every 2-edge-colouring of Kn admits a partition of its vertex set into two monochromatic cycles. It was proven for sufficiently large n by Łuczak, Rödl, and Szemerédi in 1998, later improved by Allen in 2008, and fully resolved by Bessy and Thomassé 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
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