Partitioning the hypercube into smaller hypercubes
Noga Alon, Jozsef Balogh, Vladimir N. Potapov
TL;DR
The paper studies partitions of the vertex set of the d-dimensional hypercube Q_d into disjoint subcubes, introducing f(d) and variants f_S(d) that constrain the dimensions of the parts. It relates these counts to the number of perfect matchings m(d) and m'(d) in Q_d, proving that f(d) is at most exponential in the natural parameter n relative to m(d) and showing a subexponential gap between m'(d) and m(d). It also demonstrates an exponential gap between cube-partitions with only 1- and 2-dimensional subcubes and the pure matchings, via both lower and upper bound techniques, including nibble methods and a straightforward encoding bound f(d) ≤ (d+1)^n. Additionally, the authors establish the existence of many irreducible tight subcube partitions and explore how allowing higher-dimensional subcubes influences the count, raising several open questions about the precise asymptotics. Collectively, the results connect tilings of the hypercube by subcubes to classical bounds for bipartite matchings and illuminate the combinatorial structure of these partition families.
Abstract
Denote by Q_d the d-dimensional hypercube. Addressing a recent question we estimate the number of ways the vertex set of Q_d can be partitioned into vertex disjoint smaller cubes. Among other results, we prove that the asymptotic order of this function is not much larger than the number of perfect matchings of Q_d. We also describe several new (and old) questions.