A cube dismantling problem related to bootstrap percolation

Abstract

An $n\times n\times\dots\times n$ hypercube is made from $n^d$ unit hypercubes. Two unit hypercubes are neighbours if they share a $(d-1)$-dimensional face. In each step of a dismantling process, we remove a unit hypercube that has precisely $d$ neighbours. A move is balanced if the neighbours are in $d$ orthogonal directions. In the extremal case, there are $n^{d-1}$ independent unit hypercubes left at the end of the dismantling. We call this set of hypercubes a solution. If a solution is projected in $d$ orthogonal directions and we get the entire $[n]^{d-1}$ hypercube in each direction, then the solution is perfect. We show that it is possible to use a greedy algorithm to test whether a set of hypercubes forms a solution. Perfect solutions turn out to be precisely those which can be reached using only balanced moves. Every perfect solution corresponds naturally to a Latin hypercube. However, we show that almost all Latin hypercubes do not correspond to solutions. In three dimensions, we find at least $n$ perfect solutions for every $n$, and we use our greedy algorithm to count the perfect solutions for $n\le6$. We also construct an infinite family of imperfect solutions and show that the total size of its three orthogonal projections is asymptotic to the minimum possible value. Our results solve several conjectures posed in a proceedings paper by Bar\'at, Korondi and Varga. If our dismantling process is reversed we get a build-up process very closely related to well-studied models of bootstrap percolation. We show that in an important special case our build-up reaches the same maximal position as bootstrap percolation.