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Sibirskii Matematicheskii Zhurnal, 2015, Volume 56, Number 4, Pages 775–789
DOI: https://doi.org/10.17377/smzh.2015.56.405 (Mi smj2677) 		 
This article is cited in 5 scientific papers (total in 5 papers)

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

O. V. Borodina, A. O. Ivanovab

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
b Ammosov North-Eastern Federal University, Yakutsk, Russia
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DOI: https://doi.org/10.17377/smzh.2015.56.405
Abstract: Let $\varphi_P(C_7)$ ($\varphi_T(C_7)$) 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 $15\le\varphi_T(C_7)\le17$. It is also known due to Madaras, Škrekovski, and Voss (2007) that $\varphi_P(C_7)\le359$.
We prove that $\varphi_P(C_7)=\varphi_T(C_7)=15$, which answers a question of Jendrol' et al. (1999).
Keywords: plane graph, structural properties, $3$-polytope, height.
Received: 16.11.2014
English version:
Siberian Mathematical Journal, 2015, Volume 56, Issue 4, Pages 612–623
DOI: https://doi.org/10.1134/S0037446615040059
Bibliographic databases:
Document Type: Article
UDC: 519.17
Language: Russian
Citation: O. V. Borodin, A. O. Ivanova, “Each $3$-polytope with minimum degree $5$ has a $7$-cycle with maximum degree at most $15$”, Sibirsk. Mat. Zh., 56:4 (2015), 775–789; Siberian Math. J., 56:4 (2015), 612–623
Citation in format AMSBIB
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