On spanning $k$-edge-colorable subgraphs

A subgraph H of a multigraph G is called strongly spanning, if any vertex of G is not isolated in H , while it is called maximum k-edge-colorable, if H is proper k-edge-colorable and has the largest size. We introduce a graph-parameter sp(G), that coincides with the smallest k that a graph G has a strongly spanning maximum k-edge-colorable subgraph. Our first result offers some alternative definitions of sp(G). Next, we show that ∆(G) is an upper bound for sp(G), and then we characterize the class of graphs G that satisfy sp(G) = ∆(G). Finally, we prove some bounds for sp(G) that involve well-known graphtheoretic parameters.