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Combinatorial structure of faces in triangulated 3-polytopes with minimum degree 4

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Abstract

In 1940, Lebesgue proved that every 3-polytope with minimum degree at least 4 contains a 3-face for which the set of degrees of its vertices is majorized by one of the entries: (4, 4,∞), (4, 5, 19), (4, 6, 11), (4, 7, 9), (5, 5, 9), and (5, 6, 7). This description was strengthened by Borodin (2002) to (4, 4,∞), (4, 5, 17), (4, 6, 11), (4, 7, 8), (5, 5, 8), and (5, 6, 6).

For triangulations with minimum degree at least 4, Jendrol’ (1999) gave a description of faces: (4, 4,∞), (4, 5, 13), (4, 6, 17), (4, 7, 8), (5, 5, 7), and (5, 6, 6).

We obtain the following description of faces in plane triangulations (in particular, for triangulated 3-polytopes) with minimum degree at least 4 in which all parameters are best possible and are attained independently of the others: (4, 4,∞), (4, 5, 11), (4, 6, 10), (4, 7, 7), (5, 5, 7), and (5, 6, 6).

In particular, we disprove a conjecture by Jendrol’ (1999) on the combinatorial structure of faces in triangulated 3-polytopes.

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Correspondence to O. V. Borodin.

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Original Russian Text Copyright © 2014 Borodin O.V. and Ivanova A.O.

The authors were supported by the Russian Foundation for Basic Research (Grants 12-01-00631 and 12-01-00448 (the first author) and Grants 12-01-00448 and 12-01-98510 (the second author)).

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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 55, No. 1, pp. 17–24, January–February, 2014.

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Borodin, O.V., Ivanova, A.O. Combinatorial structure of faces in triangulated 3-polytopes with minimum degree 4. Sib Math J 55, 12–18 (2014). https://doi.org/10.1134/S0037446614010030

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