Irreducible Subcube Partitions

Yuval Filmus; Edward A. Hirsch; Sascha Kurz; Ferdinand Ihringer; Artur Riazanov; Alexander Smal; Marc Vinyals

doi:10.37236/11862

Irreducible Subcube Partitions

Yuval Filmus, Edward A. Hirsch, Sascha Kurz, Ferdinand Ihringer, Artur Riazanov, Alexander Smal, Marc Vinyals

Computer Science

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

1 Scopus citations

Abstract

A subcube partition is a partition of the Boolean cube {0, 1}ⁿ into subcubes. A subcube partition is irreducible if the only sub-partitions whose union is a subcube are singletons and the entire partition. A subcube partition is tight if it “mentions” all coordinates. We study extremal properties of tight irreducible subcube partitions: minimal size, minimal weight, maximal number of points, maximal size, and maximal minimum dimension. We also consider the existence of homogeneous tight irreducible subcube partitions, in which all subcubes have the same dimensions. We additionally study subcube partitions of {0, . . . , q − 1}ⁿ, and partitions of Fⁿ2 into aﬃne subspaces, in both cases focusing on the minimal size. Our constructions and computer experiments lead to several conjectures on the extremal values of the aforementioned properties.

Original language	English
Article number	3
Journal	Electronic Journal of Combinatorics
Volume	30
Issue number	3
DOIs	https://doi.org/10.37236/11862 https://doi.org/10.37236/11862
State	Published - 8 Sep 2023

ASJC Scopus subject areas

Theoretical Computer Science
Geometry and Topology
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics
Applied Mathematics

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abstract = "A subcube partition is a partition of the Boolean cube {0, 1}n into subcubes. A subcube partition is irreducible if the only sub-partitions whose union is a subcube are singletons and the entire partition. A subcube partition is tight if it “mentions” all coordinates. We study extremal properties of tight irreducible subcube partitions: minimal size, minimal weight, maximal number of points, maximal size, and maximal minimum dimension. We also consider the existence of homogeneous tight irreducible subcube partitions, in which all subcubes have the same dimensions. We additionally study subcube partitions of {0, . . . , q − 1}n, and partitions of Fn2 into aﬃne subspaces, in both cases focusing on the minimal size. Our constructions and computer experiments lead to several conjectures on the extremal values of the aforementioned properties.",

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AU - Filmus, Yuval

AU - Hirsch, Edward A.

AU - Kurz, Sascha

AU - Ihringer, Ferdinand

AU - Riazanov, Artur

AU - Smal, Alexander

AU - Vinyals, Marc

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Irreducible Subcube Partitions

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