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Mat. Sb., 2013, Volume 204, Number 10, Pages 47–90 (Mi msb8120)  
This article is cited in 4 scientific papers (total in 4 papers)

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 НШ-2519.2012.1
МД-6277.2013.1

† Author to whom correspondence should be addressed

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

Full text: PDF file (856 kB)
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English version:
Sbornik: Mathematics, 2013, 204:10, 1435–1479

Bibliographic databases:

UDC: 519.174+519.176
MSC: 05D10, 05C15, 05C35
Received: 15.03.2012 and 12.02.2013

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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    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  crossref  mathscinet  zmath
    2. A. Sagdeev, “Lower Bounds for the Chromatic Numbers of Distance Graphs with Large Girth”, Math. Notes, 101:3 (2017), 515–528  mathnet  crossref  crossref  mathscinet  isi  elib
    3. A. Sagdeev, “The Chromatic Number of Space with Forbidden Regular Simplex”, Math. Notes, 102:4 (2017), 541–546  mathnet  crossref  crossref  mathscinet  isi  elib
    4. A. Sokolov, “On the Chromatic Numbers of Rational Spaces”, Math. Notes, 103:1-2 (2018), 111–117  mathnet  crossref  crossref  mathscinet  isi  elib
    		• Математический сборник Sbornik: Mathematics (from 1967)
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