Abstract
Eulerian and bipartite graph is a dual symmetric concept in Graph theory. It is well-known that a plane graph is Eulerian if and only if its geometric dual is bipartite. In this paper, we generalize the well-known result to embedded graphs and partial duals of cellularly embedded graphs, and characterize Eulerian and even-face graph partial duals of a cellularly embedded graph by means of half-edge orientations of its medial graph.
Highlights
With the help of Jordan Curve Theorem, the necessity of this theorem can be deduced by showing that a Eulerian plane graph is proper twoface-colourable
This does not hold for cellularly embedded graphs on the other surfaces, and one can find counter examples such as a cellularly embedding of the complete graph K5 on the torus and the cellularly embedding of a cycle Cn in the real projective plane
We further study the following topics: Extending plane bipartite graphs to even-region graphs, we generalize the above classical result to embedded graphs and partial duals of cellularly embedded graphs, and characterize all Eulerian and all even-face graph partial duals of a cellularly embedded graph by using two methods
Summary
A partial dual of a cellularly embedded graph, introduced by Chmutov [3], generalizes the geometric duality. Huggett and Moffatt [8] gave an extension of the above dual symmetry between plane Eulerian and bipartite graphs to partial duality and characterized all bipartite partial duals of plane graphs by means of oriented circuits in their medial graphs. We further study the following topics: Extending plane bipartite graphs to even-region graphs, we generalize the above classical result to embedded graphs and partial duals of cellularly embedded graphs, and characterize all Eulerian and all even-face graph partial duals of a cellularly embedded graph by using two methods
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