Rainbow matchings in strongly edge-colored graphs

A rainbow matching of an edge-colored graph G is a matching in which no two edges have the same color. There have been several studies regarding the maximum size of a rainbow matching in a properly edge-colored graph G in terms of its minimum degree ? ( G ) . Wang (2011) asked whether there exists a function f such that a properly edge-colored graph G with at least f ( ? ( G ) ) vertices is guaranteed to contain a rainbow matching of size ? ( G ) . This was answered in the affirmative later: the best currently known function Lo and Tan (2014) is f ( k ) = 4 k - 4 , for k ? 4 and f ( k ) = 4 k - 3 , for k ? 3 . Afterwards, the research was focused on finding lower bounds for the size of maximum rainbow matchings in properly edge-colored graphs with fewer than 4 ? ( G ) - 4 vertices. Strong edge-coloring of a graph G is a restriction of proper edge-coloring where every color class is required to be an induced matching, instead of just being a matching. In this paper, we give lower bounds for the size of a maximum rainbow matching in a strongly edge-colored graph G in terms of ? ( G ) . We show that for a strongly edge-colored graph G , if | V ( G ) | ? 2 ? 3 ? ( G ) 4 ? , then G has a rainbow matching of size ? 3 ? ( G ) 4 ? , and if | V ( G ) | < 2 ? 3 ? ( G ) 4 ? , then G has a rainbow matching of size ? | V ( G ) | 2 ? . In addition, we prove that if G is a strongly edge-colored graph that is triangle-free, then it contains a rainbow matching of size at least ? ( G ) .