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

TL;DR

This work addresses strong embeddings of 3-connected cubic planar graphs on surfaces with non-negative Euler characteristic by translating embeddings into dual-subgraph criteria. It provides complete, surface-specific characterizations: for the projective plane, strong embeddings correspond to $K_4$ subgraphs in the dual; for the torus, to $K_{2,2,2}$ or certain $K_{2,2m}$ (with adjacency constraints); and for the Klein bottle, to $A_3,A_5,A_6$ or certain $K_{2,2m-1}$ (with non-adjacency conditions) subgraphs in the dual. The paper formalizes the strong-embedding criterion via dual facial walks, develops algorithms (implemented in GAP) to enumerate strong embeddings and, by extension, candidate cycle double covers; it also proves existence/absence results based on cyclic edge-connectivity, and demonstrates that while projective-plane embeddings are polynomial-time computable, torus and Klein bottle cases can exhibit exponential growth in the number of strong embeddings. These results lay a robust foundation for both theoretical analysis and practical computation of cycle double covers and simplicial surfaces, with implications for higher-genus generalizations and future work on orientable surfaces via Apollonian duals.

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.

Figures (12)

• Figure 1: The three graphs $A_3$ (a), $A_5$ (b) and $A_6$ (c)
• Figure 2: Segments of facial walks (blue and orange) for $e$ not twisted (a) and $e$ twisted (b)
• Figure 3: The three graphs $A_1$ (a), $A_2$ (b) and $A_4$ (c)
• Figure 4: The graph $G$ embedded on the plane (a) with twisted edges coloured red, its dual graph $G^{\ast}$ with the twisted subgraph coloured red (b) and the re-embedding of $G$ on a torus (c)
• Figure 5: A face that has exactly two adjacent twisted edges coloured red, together with its facial walk coloured blue

Theorems & Definitions (34)

• Theorem 1
• Definition 2
• Definition 3
• Theorem 4
• Example 5
• Lemma 6
• proof
• Definition 7
• Lemma 8
• proof