The Stein theorem for loopless 2-connected plane multigraphs

TL;DR

This paper extends Stein's theorem from simple plane triangulations to all loopless 2-connected plane multigraphs, providing new equivalent conditions and broadening the theorem's applicability.

Contribution

The paper generalizes Stein's theorem to loopless 2-connected plane multigraphs and establishes additional equivalent results.

Findings

Extended Stein's theorem to multigraphs

Proved new equivalent conditions for vertex partitioning

Broadened understanding of plane multigraph properties

Abstract

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.