Rainbow spanning trees in properly coloured complete graphs
Abstract
In this short note, we study pairwise edge-disjoint rainbow spanning trees in properly edge-coloured complete graphs, where a graph is rainbow if its edges have distinct colours. Brualdi and Hollingsworth conjectured that every Kn properly edge-coloured by n-1 colours has n/2 edge-disjoint rainbow spanning trees. Kaneko, Kano and Suzuki later suggested this should hold for every properly edge-coloured Kn. Improving the previous best known bound, we show that every properly edge-coloured Kn contains (n) pairwise edge-disjoint rainbow spanning trees. Independently, Pokrovskiy and Sudakov recently proved that every properly edge-coloured Kn contains (n) isomorphic pairwise edge-disjoint rainbow spanning trees.
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