Journal of Integer Sequences, Vol. 26 (2023), Article 23.6.6

Integer Sequences from Configurations in the Hausdorff Metric Geometry via Edge Covers of Bipartite Graphs

Steven Schlicker
Grand Valley State University
Department of Mathematics
1 Campus Drive
Allendale, MI 49401

Roman Vasquez
Auburn University
Department of Mathematics and Statistics
221 Parker Hall
Auburn, AL 36849

Rachel Wofford
Pacific Northwest National Laboratory
1100 Dexter Ave N.
Seattle, WA 98109


The Hausdorff metric provides a way to measure the distance between nonempty compact sets in $\mathbb{R}^N$, from which we can build a geometry of sets. This geometry is very different than the standard Euclidean geometry and provides many interesting results. In this paper we focus on line segments in this geometry, where pairs of disjoint sets $A$ and $B$ satisfying certain distance conditions have the property that there are exactly $m$ different sets on the line segment $\overline{AB}$ at every distance from $A$, where $m$ can assume many values different than one. We provide new families of sets that generate previously unrecorded integer sequences via these values of $m$ by connecting the values of $m$ to the number of edge coverings of a graph corresponding to the sets $A$ and $B$.

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(Concerned with sequences A024023 A048291 A103453 A335608 A335609 A335610 A335611 A335612 A335613 A337416 A337417 A337418 A340173 A340174 A340175 A340199 A340200 A340201 A340403 A340404 A340405 A340433 A340434 A340435 A340436 A340437 A340438 A340897 A340898 A340899 A341551 A341552 A341553 A342327 A342328 A342580 A342796 A342850 A343372 A343373 A343374 A343800.)

Received September 13 2021; revised versions received March 2 2023; June 18 2023. Published in Journal of Integer Sequences, June 19 2023.

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