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On simplices in diameter graphs in ℝ4

Mathematical Notes volume 101, pages 265–276 (2017)Cite this article

Abstract

Agraph G is a diameter graph in ℝd if its vertex set is a finite subset in ℝd of diameter 1 and edges join pairs of vertices a unit distance apart. It is shown that if a diameter graph G in ℝ4 contains the complete subgraph K on five vertices, then any triangle in G shares a vertex with K. The geometric interpretation of this statement is as follows. Given any regular unit simplex on five vertices and any regular unit triangle in ℝ4, then either the simplex and the triangle have a common vertex or the diameter of the union of their vertex sets is strictly greater than 1.

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Authors and Affiliations

  1. Moscow Institute of Physics and Technology (State University), Dolgoprudnyi, Moscow Oblast, Russia

    A. B. Kupavskii & A. A. Polyanskii

  2. École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    A. B. Kupavskii

  3. Kharkevich Institute for Information Tranmission Problems, Russian Academy of Sciences, Moscow, Russia

    A. A. Polyanskii

  4. Technion – Israel Institute of Technology, Haifa, Israel

    A. A. Polyanskii

Authors
  1. A. B. Kupavskii
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  2. A. A. Polyanskii
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Correspondence to A. B. Kupavskii.

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Original Russian Text © A. B. Kupavskii, A. A. Polyanskii, 2017, published in Matematicheskie Zametki, 2017, Vol. 101, No. 2, pp. 232–246.

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Kupavskii, A.B., Polyanskii, A.A. On simplices in diameter graphs in ℝ4 . Math Notes 101, 265–276 (2017). https://doi.org/10.1134/S000143461701031X

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