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/√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.
Citation:
A. B. Kupavskii, A. M. Raigorodskii, “Obstructions to the realization of distance graphs with large chromatic numbers on spheres of small radii”, Sb. Math., 204:10 (2013), 1435–1479
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\by A.~B.~Kupavskii, A.~M.~Raigorodskii
\paper Obstructions to the realization of distance graphs with large chromatic numbers on spheres of small radii
\jour Sb. Math.
\yr 2013
\vol 204
\issue 10
\pages 1435--1479
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Linking options:
https://www.mathnet.ru/eng/sm8120
https://doi.org/10.1070/SM2013v204n10ABEH004345
https://www.mathnet.ru/eng/sm/v204/i10/p47
This publication is cited in the following 4 articles:
A. Sokolov, “On the Chromatic Numbers of Rational Spaces”, Math. Notes, 103:1-2 (2018), 111–117
A. Sagdeev, “Lower Bounds for the Chromatic Numbers of Distance Graphs with Large Girth”, Math. Notes, 101:3 (2017), 515–528
A. Sagdeev, “The Chromatic Number of Space with Forbidden Regular Simplex”, Math. Notes, 102:4 (2017), 541–546
A. M. Raigorodskii, “Cliques and cycles in distance graphs and graphs of diameters”, Discrete Geometry and Algebraic Combinatorics, Contemporary Mathematics, 625, 2014, 93–109
Citation in format AMSBIB
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