A Fano framework for embeddings of graphs in surfaces

TL;DR

The paper develops a unifying Fano-plane framework for seven fundamental properties of cellular embeddings of finite graphs on compact surfaces by mapping each property to a point in the Fano plane and showing that allowable combinations form lines. It translates embedding properties into parity conditions on closed walks in gems and jewels, then links these to binary edge-labelings and to bidirections on medial checkerboards, enabling robust statements about orientability, bipartiteness, and Eulerian properties under various dualities. Core contributions include a complete characterization of how the seven properties interrelate (via metatheorems), explicit correspondences to medial-direction/bidirection structures, and numerous examples illustrating all allowable property combinations; the framework also extends to Eulerian properties and to single-vertex partial duals. The results unify several known findings and provide new theorems about twisted duals, bidirections, and the structure of embeddings, with potential extensions to delta-matroids and hypermaps. Overall, the work offers a systematic, parity-based methodology for understanding how embedding properties propagate through dualities and partial duals, with practical implications for graph embeddings in topological settings.

Abstract

We consider seven fundamental properties of cellular embeddings of graphs in compact surfaces, and show that each property can be associated with a point of the Fano plane $F$, in such a way that allowable combinations of properties correspond to projective subspaces of $F$. This Fano framework allows us to deduce a number of implications involving the seven properties, providing new results and unifying existing ones. For each property, we provide a correspondence between embeddings with that property and an associated structure for $4$-regular graphs, using the medial graph of the graph embedding. We apply this to characterize when a graph embedding has a twisted dual with one of the properties. For each allowable combination of properties, we show that a graph embedding with these properties exists. We investigate connections between the seven properties and three weaker `Eulerian' properties. Our proofs involve parity conditions on closed walks in an extended version of the `gem' (graph-encoded map) representation of a graph embedding.

Figures (16)

• Figure 1: Embedded graph ${\textnormal{\em G}}$ for which we will take the barycentric subdivision and gem.
• Figure 2: Barycentric subdivision of ${\textnormal{\em G}}$.
• Figure 3: The barycentric subdivision (solid) and gem (dashed) for ${\textnormal{\em G}}$.
• Figure 4: The gem of ${\textnormal{\em G}}$.
• Figure 5: Fano Framework for Graph Embeddings.

Theorems & Definitions (67)

• Lemma 3.1
• Lemma 3.2
• Theorem 3.4
• Proposition 3.5: Wilson Wilson, Ellis-Monaghan and Moffatt twisteddualEMMGraphsonSurfEMM
• Theorem 3.6
• Theorem 3.7: Lins gemslins
• Corollary 3.8
• Theorem 3.9
• proof
• Theorem 3.10