Missing edge coverings of bipartite graphs and the geometry of the Hausdorff metric

Abstract

In this paper, we examine the problem of finding the number ${k}$ of elements at a given location on the line segment between two elements in the geometry the Hausdorff metric imposes on the set ${\mathcal{H} (\mathbb{R}^{n})}$ of all nonempty compact sets in n-dimensional real space. We demonstrate that this problem is equivalent to counting the edge coverings of simple bipartite graphs. We prove the novel results that there exist no simple bipartite graphs with exactly 19 or 37 edge coverings, and hence there do not exist any two elements of ${\mathcal{H} (\mathbb{R}^{n})}$ with exactly 19 or 37 elements at a given location lying between them—although there exist pairs of elements in ${\mathcal{H} (\mathbb{R}^{n})}$ that have k elements at any given location between them for infinitely many values of k, including k from 1 to 18 and 20 to 36. This paper extends results in the geometry of the Hausdorff metric given in J. Geom. 92: 28–59 (2009). In addition to our results about counting edge coverings, we give a brief introduction to this geometry.