Bipartite partial duals and circuits in medial 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.Comment: v2: minor changes. To appear in Combinatoric

Eulerian partial duals of plane graphs

“…By the self‐duality of the semicrossing orientation, orientations of c ‐edges on the boundary of a face of G also satisfy condition 3. Remark In , Huggett and Moffatt present a sufficient condition for a partial dual of a plane graph to be Eulerian by stating that is Eulerian for a plane graph G if A is the set of d ‐edges arising from an all‐crossing direction of . Here, we claim that for any all‐crossing direction of , there is a semicrossing orientation of with the same c ‐, d ‐vertex partition.…”

“…The proof of Theorem is based on the following two theorems proved in and a well‐known lemma. Set for .…”

“…It should be noted that the three conditions are all self‐dual, since and furthermore a c ‐edge of G corresponds to a d ‐edge of under the natural bijection. In addition, an all‐crossing direction of defined in is a half‐edge orientation of such that it contains only crossing vertices and consistent edges.…”

A Rabbit Hole between Topology and Geometry

“…Note that the dual graph is the graph embeddable on the same surface where we switch the roles of vertex and face, joining two faces if they have a common edge. This dual graph depends strongly upon the embedding, so that a graph may have different dual graphs on other surfaces: see [8] for recent research on this topic.…”

“…It is well known that any planar graph with an even number of edges on each face is bipartite; see, for example, [8]. By dualizing this statement we also know that any planar graph which is Eulerian, that is, has an even valency at each vertex, has a bipartite dual.…”