PreprintarXiv licencearXiv
Resolving Pisanski's Conjecture
Abstract
We prove Pisanski's conjecture on quadrilateral embeddings of regular bipartite graphs. For integers m≥ d≥4 with 4| 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+(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≥ d with 4| d. In each case, two cyclic permutations of a connection set of size d determine all of the local rotations.
Related
- Resolving a conjecture on permutation polynomials over 𝔽2n
- Uniform Stability of Scott-Vogelius Elements on Three-Dimensional Freudenthal Meshes in Degrees Four and Five: Resolving the Farrell-Mitchell-Scott Conjecture
- The Multiplicative Persistence Conjecture: Resolving the 2-Adic Obstruction for Nonzero Even Targets
- Resolving a conjecture on quadratic APN functions and a new quadratic (n,n)-function associated to crooked functions
- One internal stabilization is enough for resolving self-referential tubes
BibTeX
@misc{ulrigg2026resolving,
title = {{Resolving Pisanski's Conjecture}},
author = {Austin Ulrigg and Alexander Metzger},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13085},
eprint = {2610.13085},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13085},
}
RIS
TY - UNPB
TI - Resolving Pisanski's Conjecture
AU - Austin Ulrigg
AU - Alexander Metzger
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.13085
UR - https://arxiv.org/abs/2610.13085
AB - We prove Pisanski's conjecture on quadrilateral embeddings of regular bipartite graphs. For integers m≥ d≥4 with 4| 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+(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≥ d with 4| d. In each case, two cyclic permutations of a connection set of size d determine all of the local rotations.
ER -