Nonextendible latin cuboids

Darryn Bryant; Nicholas J. Cavenagh; Barbara Maenhaut; Kyle Pula; Ian M. Wanless

doi:10.1137/110825534

Nonextendible latin cuboids

Darryn Bryant^*, Nicholas J. Cavenagh, Barbara Maenhaut, Kyle Pula, Ian M. Wanless

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

9 Citations (Scopus)

Abstract

We show that for all integers m ≤ 4 there exists a 2m × 2m × m latin cuboid that cannot be completed to a 2m×2m×2m latin cube. We also show that for all even m >2 there exists a (2m-1) × (2m-1) × (m-1) latin cuboid that cannot be extended to any (2m-1) × (2m-1) × m latin cuboid.

Original language	English
Pages (from-to)	239-249
Number of pages	11
Journal	SIAM Journal on Discrete Mathematics
Volume	26
Issue number	1
DOIs	https://doi.org/10.1137/110825534
Publication status	Published - 2012
Externally published	Yes

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title = "Nonextendible latin cuboids",

abstract = "We show that for all integers m ≤ 4 there exists a 2m × 2m × m latin cuboid that cannot be completed to a 2m×2m×2m latin cube. We also show that for all even m >2 there exists a (2m-1) × (2m-1) × (m-1) latin cuboid that cannot be extended to any (2m-1) × (2m-1) × m latin cuboid.",

keywords = "Latin cube, Latin cuboid, Orthogonal latin squares, Transversal",

author = "Darryn Bryant and Cavenagh, {Nicholas J.} and Barbara Maenhaut and Kyle Pula and Wanless, {Ian M.}",

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language = "English",

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T1 - Nonextendible latin cuboids

AU - Bryant, Darryn

AU - Cavenagh, Nicholas J.

AU - Maenhaut, Barbara

AU - Pula, Kyle

AU - Wanless, Ian M.

PY - 2012

Y1 - 2012

N2 - We show that for all integers m ≤ 4 there exists a 2m × 2m × m latin cuboid that cannot be completed to a 2m×2m×2m latin cube. We also show that for all even m >2 there exists a (2m-1) × (2m-1) × (m-1) latin cuboid that cannot be extended to any (2m-1) × (2m-1) × m latin cuboid.

AB - We show that for all integers m ≤ 4 there exists a 2m × 2m × m latin cuboid that cannot be completed to a 2m×2m×2m latin cube. We also show that for all even m >2 there exists a (2m-1) × (2m-1) × (m-1) latin cuboid that cannot be extended to any (2m-1) × (2m-1) × m latin cuboid.

KW - Latin cube

KW - Latin cuboid

KW - Orthogonal latin squares

KW - Transversal

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U2 - 10.1137/110825534

DO - 10.1137/110825534

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VL - 26

SP - 239

EP - 249

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

IS - 1

ER -

Nonextendible latin cuboids

Abstract

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