Bounded Degree Spanners of the Hypercube

Rajko Nenadov,Mehtab Sawhney,Benny Sudakov,Adam Zsolt Wagner

doi:10.37236/9074

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 $\Delta_{k,\infty}(n)$ 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.

Highlights

Let Qn denote the hypercube graph, with vertex set {0, 1}n with edges connecting two vertices if they differ in precisely one coordinate
Arizumi–Hamburger–Kostochka [4] denoted by ∆k,∞(n) the minimum possible maximum degree of a k-additive spanner of Qn
Define H1 to be subgraph created by including all edges in directions in B0 for every vertex in Qn

Summary

Introduction

Let Qn denote the hypercube graph, with vertex set {0, 1}n with edges connecting two vertices if they differ in precisely one coordinate. There exists a spanning subgraph G of Qn with maximum degree at most 120 such that the diameter of G is n. Constructions of additive spanners with few edges and/or low maximum degree have attracted considerable attention in computer science in the past. Arizumi–Hamburger–Kostochka [4] denoted by ∆k,∞(n) the minimum possible maximum degree of a k-additive spanner of Qn. Note that since Qn is bipartite, by deleting edges the distance can only grow by an even amount. They showed that for k 2 and n 21 we have n e−4k ln n Their lower bound is a short argument given by counting the vertices of a certain distance from a fixed vertex, and their upper bound is an explicit construction.

Bounded degree subgraph preserving diameter

Concluding remarks and open questions