Sudoku Analogues of Baranyai's Theorem
Abstract
Motivated by higher-dimensional generalizations of Sudoku, we study exact block-structured decompositions, algebraic characterizations, and orthogonality for Sudoku hypercubes. Let n=∏i=1d ai, let bi=n/ai, and consider the λ-fold complete d-uniform d-partite hypergraph with d vertex classes of size n, where the ith class is partitioned into ai groups of size bi. Given positive integers m1,…,mk with ∑j=1k mj=λnd, we partition the edges into color classes of sizes m1,…,mk so that, in color j, vertex degrees and block counts are each either ⌊ mj/n⌋ or ⌈ mj/n⌉, while the multiplicity of an underlying edge is either ⌊ mj/nd⌋ or ⌈ mj/nd⌉. When mj=nrj, the vertex and block balances are exact, yielding block factorizations and higher-dimensional Sudoku analogues of Baranyai's theorem. Within the same block framework, we give a Delsarte characterization of the Sudoku condition using association schemes and study mutually orthogonal Sudoku hypercubes of order q3 for prime powers q. For block sizes (q3,q2,q) and (q3,q3,1), the resulting families attain a general upper bound and are best possible. For block size (q2,q2,q2), we construct q2(q2−1)(q2−q) mutually orthogonal hypercubes; this construction is asymptotically best possible as q→∞.
Related
- Analogues of Sylow's first theorem, Cauchy's theorem, and Hall's theorem for skew braces
- Analogues of Grün's lemma and Baer's theorem for skew left braces
- On Non-Uniform Baranyai Factorizations
- Measuring Sudoku Difficulty: Formulation, Propagation and Uniqueness Certification
- q-Analogues of some supercongruences related to Van Hamme's (E.2) and (F.2) supercongruences
BibTeX
@misc{bahmanian2026sudoku,
title = {{Sudoku Analogues of Baranyai's Theorem}},
author = {Amin Bahmanian and Sho Suda},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23975},
eprint = {2609.23975},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23975},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sudoku Analogues of Baranyai's Theorem
AU - Amin Bahmanian
AU - Sho Suda
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.23975
UR - https://arxiv.org/abs/2609.23975
AB - Motivated by higher-dimensional generalizations of Sudoku, we study exact block-structured decompositions, algebraic characterizations, and orthogonality for Sudoku hypercubes. Let n=∏_i=1^d a_i, let b_i=n/a_i, and consider the λ-fold complete d-uniform d-partite hypergraph with d vertex classes of size n, where the ith class is partitioned into a_i groups of size b_i. Given positive integers m_1,…,m_k with ∑_j=1^k m_j=λn^d, we partition the edges into color classes of sizes m_1,…,m_k so that, in color j, vertex degrees and block counts are each either ⌊ m_j/n⌋ or ⌈ m_j/n⌉, while the multiplicity of an underlying edge is either ⌊ m_j/n^d⌋ or ⌈ m_j/n^d⌉. When m_j=nr_j, the vertex and block balances are exact, yielding block factorizations and higher-dimensional Sudoku analogues of Baranyai's theorem. Within the same block framework, we give a Delsarte characterization of the Sudoku condition using association schemes and study mutually orthogonal Sudoku hypercubes of order q^3 for prime powers q. For block sizes (q^3,q^2,q) and (q^3,q^3,1), the resulting families attain a general upper bound and are best possible. For block size (q^2,q^2,q^2), we construct q^2(q^2−1)(q^2−q) mutually orthogonal hypercubes; this construction is asymptotically best possible as q→∞.
ER -