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arXiv · 2608.17996

A rainbow version of Lehel's conjecture

Pedro Araújo·Xiao-Chuan Liu·Taísa Martins·Walner Mendonça·Luiz Moreira·Vinícius Fernandes dos Santos·Zhifei Yan
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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

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Pedro Araújo, Xiao-Chuan Liu, Taísa Martins, Walner Mendonça, Luiz Moreira, Vinícius Fernandes dos Santos, Zhifei Yan. 2026-08-18. A rainbow version of Lehel's conjecture. https://arxiv.org/abs/2608.17996

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