Existences of rainbow matchings and rainbow matching covers

Abstract

Let G be an edge-coloured graph. A rainbow subgraph in G is a subgraph such that its edges have distinct colours. The minimum colour degree δc(G) of G is the smallest number of distinct colours on the edges incident with a vertex of G. We show that every edge-coloured graph G on n≥7k/2+2 vertices with δc(G)≥k contains a rainbow matching of size at least k, which improves the previous result for k≥10.Let Δmon(G) be the maximum number of edges of the same colour incident with a vertex of G. We also prove that if t≥11 and Δmon(G)≤t, then G can be edge-decomposed into at most ⌊tn/2⌋ rainbow matchings. This result is sharp and improves a result of LeSaulnier and West.