2022/12/30 by Filmus, Yuval, Edward Hirsch, Ferdinand Ihringer +10
Computer Science · #004 #510 #Advanced Graph Theory Research #Boolean #Complexity and Algorithms in Graphs #Interconnection Networks and Systems #hitting formulas #hypercubes #partitions
paper · pdf · doi:10.15495/epub_ubt_00006837
openalex publication_date 2022/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
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 \mathbbF2n 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.