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 the restriction k 4, but Li and Xu proved the conjecture for all other properly coloured complete graphs. LeSaulnier, Stocker, Wenger and West showed that a rainbow matching of size k/2 is guaranteed to exist, and they proved several sufficient conditions for a matching of size k/2 . We prove the conjecture in full.