Structural characterization of bipartite graphs admitting orientable quadrangulations

Characterize the structural conditions that guarantee a connected regular bipartite graph admits an orientable quadrangulation, recognizing that the necessary numerical conditions are not sufficient for every prescribed graph.

Background

The main theorem establishes that, whenever m≥d≥4 and 4 divides md, one can choose a connected simple d-regular bipartite graph with m vertices in each bipartition class that admits an orientable quadrangulation. The result does not imply that every graph with these parameters has such an embedding.

The paper notes that Pisanski constructed a connected 4-regular bipartite graph with nine vertices in each bipartition class and girth four that has no orientable quadrangulation. Thus, the unresolved issue is to identify graph-theoretic or structural criteria distinguishing prescribed graphs that do and do not admit orientable quadrangulations.