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) |
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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. |
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ISSN: | 1077-8926 1077-8926 |
DOI: | 10.37236/9074 |