Year: 2026
Type: preprint
Abstract: We prove Pisanski's conjecture on quadrilateral embeddings of regular bipartite graphs. For integers $m\geq d\geq4$ with $4\mid md$, we construct both a connected simple $d$-regular bipartite graph $G$ with $m$ vertices in each bipartition class and an orientable quadrangulation. This graph's genus is \[ \ga(G)=1+\frac{m(d-4)}4. \] These quadrangulations attain the lower bound on the genus from Euler's formula. We give dihedral constructions for even $m$ and a cyclic construction for every $m\geq d$ with $4\mid d$. In each case, two cyclic permutations of a connection set of size $d$ determine all of the local rotations.
Source: arXiv (Cornell University)
Authors: Ulrigg, Austin, Metzger, Alexander
Institutions: University of Washington, University of California, Berkeley
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W7221577814