A Rose-Coloured Rainbow

Rainbow matchings in properly-coloured multigraphs

Aharoni and Berger conjectured that in any bipartite multigraph that is properly edge-coloured by n colours with at least n + 1 edges of each colour there must be a matching that uses each colour exactly once. In this paper we consider the same question without the bipartiteness assumption. We show that in any multigraph with edge multiplicities o(n) that is properly edge-coloured by n colours ...

Rainbow matchings and connectedness of coloured graphs

Aharoni and Berger conjectured that every bipartite graph which is the union of n matchings of size n + 1 contains a rainbow matching of size n. This conjecture is a generalization of several old conjectures of Ryser, Brualdi, and Stein about transversals in Latin squares. When the matchings are all edge-disjoint and perfect, an approximate version of this conjecture follows from a theorem of H...

Large Rainbow Matchings in Edge-Coloured Graphs

A rainbow subgraph of an edge-coloured graph is a subgraph whose edges have distinct colours. The colour degree of a vertex v is the number of different colours on edges incident with v. Wang and Li conjectured that for k 4, every edge-coloured graph with minimum colour degree k contains a rainbow matching of size at least k/2 . A properly edge-coloured K4 has no such matching, which motivates ...

Rainbow spanning trees in properly coloured complete graphs

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-c...

[Rose is a rose is a rose].