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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
USA
schlicks@gvsu.edu

    

Roman Vasquez
Auburn University
Department of Mathematics and Statistics
221 Parker Hall
Auburn, AL 36849
USA
roman.vasquez@auburn.edu

Rachel Wofford
Pacific Northwest National Laboratory
1100 Dexter Ave N.
Seattle, WA 98109
USA
re.wofford@pnnl.gov

in

Abstract:

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 $B$.