Strong Embeddings of 3-Connected Cubic Planar Graphs on Surfaces of non-negative Euler Characteristic

TL;DR

This paper characterizes strong embeddings of 3-connected cubic planar graphs on non-spherical surfaces with non-negative Euler characteristic, linking them to subgraph conditions and advancing understanding of cycle double covers.

Contribution

It provides a complete characterization of strong embeddings on the projective plane, torus, and Klein bottle, connecting structural properties to subgraph criteria.

Findings

Characterization of strong embeddings on the projective plane, torus, and Klein bottle.

Explicit criteria for the non-existence of strong embeddings.

Foundations for computing cycle double covers on these surfaces.

Abstract

Whitney proved that 3-connected planar graphs admit a unique embedding on the sphere. In contrast, Enami investigated embeddings of 3-connected cubic planar graphs on non-spherical surfaces with non-negative Euler characteristic. He established that such an embedding exists if and only if the dual graph contains a particular subgraph. Here, strong embeddings are investigated motivated by the cycle double cover conjecture and the relation to triangulated surfaces. We provide a complete characterization of strong embeddings on the projective plane, the torus, and the Klein bottle in terms of a distinguished subset of Enami's subgraphs. This characterization not only deepens the structural understanding of graph embeddings on non-spherical surfaces, but also establishes a robust foundation for computing cycle double covers. As a direct consequence, we derive explicit criteria that determine when a graph does not admit a strong embedding on these surfaces-offering new tools for both theoretical analysis and algorithmic applications.