An enhancement of Nash–Williams' Theorem on edge arboricity of graphs

The well-known Nash–Williams' Theorem states that for any positive integer $k$ a multigraph $G=(V,E)$ admits an edge decomposition into $k$ forests iff every subset $X\subseteq V$ induces a subgraph $G[X]$ with at most $k(|X|-1)$ edges. In this paper we prove that, under certain conditions, this decomposition can be chosen so that each forest contains no isolated vertices. More precisely, we prove that if either $G$ is a bipartite multigraph with minimum degree $\delta(G)\ge k$, or $k=2$ and $\delta(G)\ge 3$, then $G$ can be decomposed into $k$ forests without isolated vertices.