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Each 3-polytope with minimum degree 5 has a 7-cycle with maximum degree at most 15

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Abstract

Let φP (C 7) ( φT (C 7)) be the minimum integer k with the property that each 3-polytope (respectively, each plane triangulation) with minimum degree 5 has a 7-cycle with all vertices of degree at most k. In 1999, Jendrol’, Madaras, Soták, and Tuza proved that 15 ≤ φT (C 7) ≤ 17. It is also known due to Madaras, φSkrekovski, and Voss (2007) that φP (C 7) ≤ 359.

We prove that φP (C 7) = φT (C 7) = 15, which answers a question of Jendrol’ et al. (1999).

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    O. V. Borodin

  2. Ammosov North-Eastern Federal University, Yakutsk, Russia

    A. O. Ivanova

Authors
  1. O. V. Borodin
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  2. A. O. Ivanova
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Corresponding author

Correspondence to O. V. Borodin.

Additional information

Original Russian Text Copyright © 2015 Borodin O.V. and Ivanova A.O.

The authors were supported by the Russian Foundation for Basic Research (Grants 12–01–00631 and 15–01–05867 for the first author and Grant 12-01-98510 for the second author) and the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-1939.2014.1). The second author worked within the governmental task “Organization of Scientific Research.”

Novosibirsk; Yakutsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 56, No. 4, pp. 775–789, July–August, 2015; DOI: 10.17377/smzh.2015.56.405.

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Borodin, O.V., Ivanova, A.O. Each 3-polytope with minimum degree 5 has a 7-cycle with maximum degree at most 15. Sib Math J 56, 612–623 (2015). https://doi.org/10.1134/S0037446615040059

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  • DOI: https://doi.org/10.1134/S0037446615040059

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