Bipartite partial duals and circuits in medial graphs

TL;DR

This paper extends the classical Eulerian-bipartite duality to partial duals of plane graphs and characterizes bipartite partial duals using oriented circuits in medial graphs.

Contribution

It introduces a characterization of bipartite partial duals of plane graphs via oriented circuits in their medial graphs, generalizing known duality results.

Findings

Bipartite partial duals are characterized by oriented circuits in medial graphs.

Extension of Eulerian-bipartite duality to partial duals of plane graphs.

Provides a new framework for understanding graph dualities in planar graphs.

Abstract

It is well known that a plane graph is Eulerian if and only if its geometric dual is bipartite. We extend this result to partial duals of plane graphs. We then characterize all bipartite partial duals of a plane graph in terms of oriented circuits in its medial graph.