A Fano framework for embeddings of graphs in surfaces

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.