On strongly 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 $\Delta(G)$ is an upper bound for $sp(G)$, and then we characterize the class of graphs $G$ that satisfy $sp(G)=\Delta(G)$. Finally, we prove some bounds for $sp(G)$ that involve well-known graph-theoretic parameters.