Bounded Degree Spanners of the Hypercube

 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 giv...

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Veröffentlicht in:The Electronic journal of combinatorics 2020-07, Vol.27 (3)
Hauptverfasser: Nenadov, Rajko, Sawhney, Mehtab, Sudakov, Benny, Wagner, Adam Zsolt
Format: Artikel
Sprache:eng
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Zusammenfassung: 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.
ISSN:1077-8926
1077-8926
DOI:10.37236/9074