The Stein theorem for loopless 2-connected plane multigraphs

Stein proved that for each simple plane triangulation H there exists a partitioning of the vertex of H into two subsets each of which induces a forest if and only if the dual H^{*} has a Hamilton cycle. We extend the Stein theorem for graphs in the family of all loopless 2-connected plane multigraphs and we prove some other equivalent results.