Existences of rainbow matchings and rainbow matching covers

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 incidentwith a vertex ofG.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∆mon(G) be themaximumnumber of edges of the same colour incidentwith a vertex of G. We also prove that if t ≥ 11 and ∆mon(G) ≤ t , then G can be edge-decomposed into at most ⌊tn/2⌋ rainbowmatchings. This result is sharp and improves a result of LeSaulnier and West. © 2015 The Author. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).