2017/04/24 by Balogh, József, Liu, Hong, Montgomery, Richard
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1704.07200
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 Ω(n) pairwise edge-disjoint rainbow spanning trees. Independently, Pokrovskiy and Sudakov recently proved that every properly edge-coloured Kn contains Ω(n) isomorphic pairwise edge-disjoint rainbow spanning trees.