Bounded Degree Spanners of the Hypercube

TL;DR

This paper constructs bounded degree subgraphs of hypercubes with diameter n and improves bounds on the minimum degree of k-additive spanners, advancing understanding of hypercube subgraph properties.

Contribution

Provides an explicit construction of a bounded degree hypercube subgraph with diameter n and refines upper bounds on k-additive spanner degrees.

Findings

Constructed a hypercube subgraph with maximum degree at most 120 and diameter n.

Improved the upper bound on the minimum degree of k-additive spanners to involve iterated logarithms.

Enhanced theoretical bounds on hypercube subgraph properties.

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 Erd\H{o}s--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 $\Delta_{k,\infty}(n)$ the minimum possible maximum degree of a $k$-additive spanner of $Q_n$, Arizumi--Hamburger--Kostochka showed that $$\frac{n}{\ln n}e^{-4k}\leq \Delta_{2k,\infty}(n)\leq 20\frac{n}{\ln n}\ln \ln n.$$ We improve their upper bound by showing that $$\Delta_{2k,\infty}(n)\leq 10^{4k} \frac{n}{\ln n}\ln^{(k+1)}n,$$where the last term denotes a $k+1$-fold iterated logarithm.