Almost all optimally coloured complete graphs contain a rainbow Hamilton path
Journal of Combinatorial Theory, Series B(2022)
Abstract
A subgraph H of an edge-coloured graph is called rainbow if all of the edges of H have different colours. In 1989, Andersen conjectured that every proper edge-colouring of Kn admits a rainbow path of length n−2. We show that almost all optimal edge-colourings of Kn admit both (i) a rainbow Hamilton path and (ii) a rainbow cycle using all of the colours. This result demonstrates that Andersen's Conjecture holds for almost all optimal edge-colourings of Kn and answers a recent question of Ferber, Jain, and Sudakov. Our result also has applications to the existence of transversals in random symmetric Latin squares.
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Key words
Andersen's conjecture,Edge-colouring,Rainbow path,Rainbow cycle,Hypergraph matchings,Distributive absorption,Latin squares
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