Extending Symmetric Layer-Rainbow Latin Cubes

Abstract

An n× n× n array on n² symbols is a layer-rainbow Latin cube if every layer contains every symbol exactly once. We call it symmetric if L_ijℓ=L_{jℓ i}=L_{ℓ ij} for distinct i,j,ℓ and L_iij=L_jji, L_iji=L_jij, L_ijj=L_jii for distinct i,j. We determine exactly when a symmetric layer-rainbow Latin cube of order m embeds in one of order n, giving a three-dimensional analogue of Cruse's embedding theorem. Call a positive integer admissible if it is congruent to $0$ or $2$ modulo $3$, with $1$ admissible and $3$ excluded. For n>m, an embedding exists if and only if m,n are admissible, (m,n)≠(2,5), and n≥2m,&n−mnot≡1pmod3, [1mm] n≥ m+(√(48m²+1)−1)/6,&n−m≡1pmod3. Via the equivalent one-factorization problem for a non-uniform hypergraph, fair detachment reduces the proof to an exact integer allocation. We also determine the structure forced at both sharp boundaries and obtain infinitely many equality cases. At order eight, we construct a symmetric layer-rainbow Latin cube admitting the natural diagonal action of PSL(2,7), whose induced action on the $64$ symbols has orbit sizes $1,7,28,28$.

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BibTeX RIS

BibTeX

@misc{bahmanian2026extending,
  title = {{Extending Symmetric Layer-Rainbow Latin Cubes}},
  author = {Amin Bahmanian},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.09621},
  eprint = {2609.09621},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.09621},
}

RIS

TY  - UNPB
TI  - Extending Symmetric Layer-Rainbow Latin Cubes
AU  - Amin Bahmanian
PY  - 2026
DA  - 2026/09/09
PB  - arXiv
DO  - 10.48550/arxiv.2609.09621
UR  - https://arxiv.org/abs/2609.09621
AB  - An n× n× n array on n^2 symbols is a layer-rainbow Latin cube if every layer contains every symbol exactly once. We call it symmetric if L_ijℓ=L_jℓ i=L_ℓ ij for distinct i,j,ℓ and L_iij=L_jji, L_iji=L_jij, L_ijj=L_jii for distinct i,j. We determine exactly when a symmetric layer-rainbow Latin cube of order m embeds in one of order n, giving a three-dimensional analogue of Cruse's embedding theorem. Call a positive integer admissible if it is congruent to $0$ or $2$ modulo $3$, with $1$ admissible and $3$ excluded. For n>m, an embedding exists if and only if m,n are admissible, (m,n)≠(2,5), and n≥2m,&n−mnot≡1pmod3, [1mm] n≥ m+(√(48m^2+1)−1)/6,&n−m≡1pmod3. Via the equivalent one-factorization problem for a non-uniform hypergraph, fair detachment reduces the proof to an exact integer allocation. We also determine the structure forced at both sharp boundaries and obtain infinitely many equality cases. At order eight, we construct a symmetric layer-rainbow Latin cube admitting the natural diagonal action of PSL(2,7), whose induced action on the $64$ symbols has orbit sizes $1,7,28,28$.
ER  -

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