Bounded degree spanners of the hypercube

Rajko Nenadov ; Mehtab Sawhney ; Benny Sudakov¹ ; Adam Zsolt Wagner

¹Eidgenössische Technische Hochschule Zürich

The electronic journal of combinatorics, Tome 27 (2020) no. 3

Résumé

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.

Détail

DOI : 10.37236/9074

Classification : 05C65, 05C12, 05C60, 05C07, 68R10
Mots-clés : \(k\)-additive spanners, bounded degree subgraph