The crossing number and the unit-distance crossing number of the Hamming graphs H(d,3)=K_3^{box d}, and their realizations over many coordinate fields
Abstract
The Hamming graph H(d,3)=K_3^{box d} (n=3^d vertices) is the graph of single-symbol errors of ternary codes, yet we ask a geometric question of it: can it be drawn in the plane with every edge exactly one unit long and no two non-adjacent vertices a unit apart? It can, for every d, joining two problems into one. Where does it live? As a Minkowski sum of unit triangles, each H(d,3) has a hidden flexibility carrying its coordinates up the constructibility ladder (compass, origami, and beyond), so one graph is realizable over many number fields at once, in the plane and in R^3 (edim(H(d,q))=q-1). Yet almost every faithful realization is transcendental: the Galois picture is a measure-zero shadow of a vast transcendental continuum, matching the exists-R hardness of unit-distance recognition. How crowded must a unit drawing be? We separate the ordinary crossing number from a unit-distance crossing number (absent from Schaefer's survey), and one recursion H(d,3)=H(d-1,3) box K_3 controls both. We prove cr(H)=Theta(n^2), with sharp one-page constant 7/6; and Omega(n^2) <= udcr(H) <= O(n^2 log n) by concentration, plus a closed-form majorant (3/2)n^2(L^2-L+1), L=log_3 n. The least-crossing drawing we find is constructible (origami): field and crossings are two readings of one geometry. All claims are verified computationally; the lower bound udcr=Omega(n^2 log n) is the central open problem.
Related
BibTeX
@misc{filho2026crossing,
title = {{The crossing number and the unit-distance crossing number of the Hamming graphs H(d,3)=K_3^box d, and their realizations over many coordinate fields}},
author = {Haroldo Costa Silva Filho},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13757},
eprint = {2609.13757},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13757},
}
RIS
TY - UNPB
TI - The crossing number and the unit-distance crossing number of the Hamming graphs H(d,3)=K_3^{box d}, and their realizations over many coordinate fields
AU - Haroldo Costa Silva Filho
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13757
UR - https://arxiv.org/abs/2609.13757
AB - The Hamming graph H(d,3)=K_3^{box d} (n=3^d vertices) is the graph of single-symbol errors of ternary codes, yet we ask a geometric question of it: can it be drawn in the plane with every edge exactly one unit long and no two non-adjacent vertices a unit apart? It can, for every d, joining two problems into one. Where does it live? As a Minkowski sum of unit triangles, each H(d,3) has a hidden flexibility carrying its coordinates up the constructibility ladder (compass, origami, and beyond), so one graph is realizable over many number fields at once, in the plane and in R^3 (edim(H(d,q))=q-1). Yet almost every faithful realization is transcendental: the Galois picture is a measure-zero shadow of a vast transcendental continuum, matching the exists-R hardness of unit-distance recognition. How crowded must a unit drawing be? We separate the ordinary crossing number from a unit-distance crossing number (absent from Schaefer's survey), and one recursion H(d,3)=H(d-1,3) box K_3 controls both. We prove cr(H)=Theta(n^2), with sharp one-page constant 7/6; and Omega(n^2) <= udcr(H) <= O(n^2 log n) by concentration, plus a closed-form majorant (3/2)n^2(L^2-L+1), L=log_3 n. The least-crossing drawing we find is constructible (origami): field and crossings are two readings of one geometry. All claims are verified computationally; the lower bound udcr=Omega(n^2 log n) is the central open problem.
ER -