An Extremal Problem in the Hypercube and Optimization of Asynchronous Circuits

We prove that if m ≥ 2, then the minimumk ∈ N such that the k -cube {0, 1}kcan be decomposed as the disjoint union of m connected adjacent subsets satisfies 2log2m − log2log2m − 1 ≤ k ≤ 2⌈log2m⌉ − ⌊log2log2m⌋ + 5.