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 Mat. Sb., 2013, Volume 204, Number 10, Pages 47–90 (Mi msb8120)

Obstructions to the realization of distance graphs with large chromatic numbers on spheres of small radii

A. B. Kupavskiiab, A. M. Raigorodskiiba

a Moscow Institute of Physics and Technology
b M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We investigate in detail some properties of distance graphs constructed on the integer lattice. Such graphs find wide applications in problems of combinatorial geometry, in particular, such graphs were employed to answer Borsuk's question in the negative and to obtain exponential estimates for the chromatic number of the space.
This work is devoted to the study of the number of cliques and the chromatic number of such graphs under certain conditions. Constructions of sequences of distance graphs are given, in which the graphs have unit length edges and contain a large number of triangles that lie on a sphere of radius $1/\sqrt{3}$ (which is the minimum possible). At the same time, the chromatic numbers of the graphs depend exponentially on their dimension. The results of this work strengthen and generalize some of the results obtained in a series of papers devoted to related issues.
Bibliography: 29 titles.

Keywords: distance graph, chromatic number, clique, sphere of smallest radius.

 Funding Agency Grant Number Russian Foundation for Basic Research 12-01-00683 Ministry of Education and Science of the Russian Federation ÍØ-2519.2012.1ÌÄ-6277.2013.1

Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm8120

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English version:
Sbornik: Mathematics, 2013, 204:10, 1435–1479

Bibliographic databases:

UDC: 519.174+519.176
MSC: 05D10, 05C15, 05C35

Citation: A. B. Kupavskii, A. M. Raigorodskii, “Obstructions to the realization of distance graphs with large chromatic numbers on spheres of small radii”, Mat. Sb., 204:10 (2013), 47–90; Sb. Math., 204:10 (2013), 1435–1479

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb8120
• https://doi.org/10.4213/sm8120
• http://mi.mathnet.ru/eng/msb/v204/i10/p47

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. M. Raigorodskii, “Cliques and cycles in distance graphs and graphs of diameters”, Discrete Geometry and Algebraic Combinatorics, Contemporary Mathematics, 625, 2014, 93–109
2. A. Sagdeev, “Lower Bounds for the Chromatic Numbers of Distance Graphs with Large Girth”, Math. Notes, 101:3 (2017), 515–528
3. A. Sagdeev, “The Chromatic Number of Space with Forbidden Regular Simplex”, Math. Notes, 102:4 (2017), 541–546
4. A. Sokolov, “On the Chromatic Numbers of Rational Spaces”, Math. Notes, 103:1-2 (2018), 111–117
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