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Coloring Distance Graphs and Graphs of Diameters

In: Thirty Essays on Geometric Graph Theory

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  • Andrei M. Raigorodskii

    (Moscow State University, Mechanics and Mathematics Faculty, Department of Mathematical Statistics and Random Processes
    Moscow Institute of Physics and Technology, Faculty of Innovations and High Technology, Department of Data Analysis
    Yandex Research Laboratories)

Abstract

In this chapter, we discuss two classical problems lying on the edge of graph theory and combinatorial geometry. The first problem is due to E. Nelson. It consists of coloring metric spaces in such a way that pairs of points at some prescribed distances receive different colors. The second problem is attributed to K. Borsuk and involves finding the minimum number of parts of smaller diameter into which an arbitrary bounded nonsingleton point set in a metric space can be partitioned. Both problems are easily translated into the language of graph theory, provided we consider, instead of the whole space, any (finite) distance graph or any (finite) graph of diameters. During the last decades, a huge number of ideas have been proposed for solving both problems, and many results in both directions have been obtained. In the survey below, we try to give an entire picture of this beautiful area of geometric combinatorics.

Suggested Citation

  • Andrei M. Raigorodskii, 2013. "Coloring Distance Graphs and Graphs of Diameters," Springer Books, in: János Pach (ed.), Thirty Essays on Geometric Graph Theory, pages 429-460, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4614-0110-0_23
    DOI: 10.1007/978-1-4614-0110-0_23
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