Bounded Degree Spanners of the Hypercube

Abstract

In this short note we study two questions about the existence of subgraphs of the hypercube Q n with certain properties. The first question, due to Erdős-Hamburger-Pippert-Weakley, asks whether there exists a bounded degree subgraph of Q n which has diameter n. We answer this question by giving an explicit construction of such a subgraph with maximum degree at most 120.The second problem concerns properties of k-additive spanners of the hypercube, that is, subgraphs of Q n in which the distance between any two vertices is at most k larger than in Q n . Denoting by ∆ k,∞ (n) the minimum possible maximum degree of a k-additive spanner of Q n , Arizumi-Hamburger-Kostochka showed that n ln n e −4k ≤ ∆ 2k,∞ (n) ≤ 20 n ln n ln ln n.We improve their upper bound by showing thatwhere the last term denotes a k + 1-fold iterated logarithm.