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Mat. Zametki, 2015, Volume 97, Issue 3, Pages 342–349 (Mi mz10550)  

This article is cited in 4 scientific papers (total in 4 papers)

New Upper Bound for the Chromatic Numberof a Random Subgraph of a Distance Graph

A. S. Gusev


Abstract: This paper is related to the classical Hadwiger–Nelson problem dealing with the chromatic numbers of distance graphs in ${\mathbb R}^n$. We consider the class consisting of the graphs $G(n,2s+1,s)=(V(n,2s+1), E(n,2s+1,s))$ defined as follows:
\begin{align*} V(n,2s+1)&=\{x=(x_1,x_2,…,x_n): x_i\in \{0,1\},   x_1+x_2+…+x_n=2s+1\}, E(n,2s+1,s)&=\{\{x,y\}:(x,y)=s\}, \end{align*}
where $(x,y)$ stands for the inner product. We study the random graph ${\mathcal G}(G(n,2s+1,s),p)$ each of whose edges is taken from the set $E(n,2s+1,s)$ with probability $p$ independently of the other edges. We prove a new bound for the chromatic number of such a graph.

Keywords: Hadwiger–Nelson problem, distance graph, random subgraph, chromatic number, Turán number.

DOI: https://doi.org/10.4213/mzm10550

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English version:
Mathematical Notes, 2015, 97:3, 326–332

Bibliographic databases:

UDC: 519.174
Received: 10.08.2014

Citation: A. S. Gusev, “New Upper Bound for the Chromatic Numberof a Random Subgraph of a Distance Graph”, Mat. Zametki, 97:3 (2015), 342–349; Math. Notes, 97:3 (2015), 326–332

Citation in format AMSBIB
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\by A.~S.~Gusev
\paper New Upper Bound for the Chromatic Number\\ of a~Random Subgraph of a~Distance Graph
\jour Mat. Zametki
\yr 2015
\vol 97
\issue 3
\pages 342--349
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\transl
\jour Math. Notes
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\vol 97
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\pages 326--332
\crossref{https://doi.org/10.1134/S0001434615030037}
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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, “_orig Combinatorial Geometry and Coding Theory”, Fundam. Inform., 145:3 (2016), 359–369  crossref  mathscinet  zmath  isi  elib  scopus
    2. S. G. Kiselev, A. M. Raigorodskii, “On the chromatic number of a random subgraph of the Kneser graph”, Dokl. Math., 96:2 (2017), 475–476  crossref  crossref  mathscinet  zmath  isi  elib  scopus
    3. D. D. Cherkashin, A. M. Raigorodskii, “On the chromatic numbers of low-dimensional spaces”, Dokl. Math., 95:1 (2017), 5–6  crossref  crossref  mathscinet  zmath  isi  elib  scopus
    4. A. S. Gusev, “Clique numbers of random subgraphs of some distance graphs”, Problems Inform. Transmission, 54:2 (2018), 165–175  mathnet  crossref  isi  elib
  • Математические заметки Mathematical Notes
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