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Sbornik: Mathematics, 2013, Volume 204, Issue 10, Pages 1435–1479
DOI: https://doi.org/10.1070/SM2013v204n10ABEH004345 (Mi sm8120) 		 
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
English version PDF (795 kB) Citations (4)
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DOI: https://doi.org/10.1070/SM2013v204n10ABEH004345
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
Received: 15.03.2012 and 12.02.2013
Russian version:
Matematicheskii Sbornik, 2013, Volume 204, Number 10, Pages 47–90
DOI: https://doi.org/10.4213/sm8120
Bibliographic databases:
Document Type: Article
UDC: 519.174+519.176
MSC: 05D10, 05C15, 05C35
Language: English
Original paper language: Russian
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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    • This publication is cited in the following 4 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    References:61
    First page:40
     
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