Abstract
The height of a face in a 3-polytope is the maximum degree of the incident vertices of the face, and the height of a 3-polytope, h, is the minimum height of its faces. A face is pyramidal if it is either a 4-face incident with three 3-vertices, or a 3-face incident with two vertices of degree at most 4. If pyramidal faces are allowed, then h can be arbitrarily large; so we assume the absence of pyramidal faces. In 1940, Lebesgue proved that every quadrangulated 3-polytope has h ≤ 11. In 1995, this bound was lowered by Avgustinovich and Borodin to 10. Recently, we improved it to the sharp bound 8. For plane triangulation without 4-vertices, Borodin (1992), confirming the Kotzig conjecture of 1979, proved that h ≤ 20 which bound is sharp. Later, Borodin (1998) proved that h ≤ 20 for all triangulated 3-polytopes. Recently, we obtained the sharp bound 10 for triangle-free 3-polytopes. In 1996, Horňák and Jendrol’ proved for arbitrarily 3-polytopes that h ≤ 23. In this paper we improve this bound to the sharp bound 20.
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References
Steinitz E., “Polyeder und Raumeinteilungen,” Enzykl. Math. Wiss. (Geometrie), vol. 3AB, no. 12, 1–139 (1922).
Lebesgue H., “Quelques conséquences simples de la formule d’Euler,” J. Math. Pures Appl., vol. 19, 27–43 (1940).
Borodin O. V., “Colorings of plane graphs: a survey,” Discrete Math., vol. 313, no. 4, 517–539 (2013).
Ore O. and Plummer M. D., “Cyclic coloration of plane graphs,” in: Recent Progress in Combinatorics (W. T. Tutte, Ed.), Academic Press, New York, 1969, 287–293.
Borodin O. V., “Strengthening Lebesgue’s theorem on the structure of the minor faces in convex polyhedra,” Diskretn. Anal. Issled. Oper. Ser. 1, vol. 9, no. 3, 29–39 (2002).
Borodin O. V., “Solving the Kotzig and Grünbaum problems on the separability of a cycle in planar graphs,” Mat. Zametki, vol. 46, no. 5, 9–12 (1989).
Borodin O. V. and Ivanova A. O., “Describing 3-faces in normal plane maps with minimum degree 4,” Discrete Math., vol. 313, no. 23, 2841–2847 (2013).
Borodin O. V., Ivanova A. O., and Kostochka A. V., “Describing faces in plane triangulations,” Discrete Math., vol. 319, 47–61 (2014).
Kotzig A., “From the theory of Eulerian polyhedrons,” Mat. Eas. SAV (Math. Slovaca), vol. 13, 20–34 (1963).
Grünbaum B., “Polytopal graphs,” in: Studies in Graph Theory. Part II (Ed. D. R. Fulkerson), Washington, D. C., Math. Assoc. Amer., vol. 12, 1975, 201–224 (MAA Stud. Math.).
Plummer M. D., “On the cyclic connectivity of planar graphs,” in: Graph Theory and Applications, Springer-Verlag, Berlin, 1972, 235–242.
Kotzig A., “Extremal polyhedral graphs,” Ann. New York Acad. Sci., vol. 319, 569–570 (1979).
Borodin O. V., “Minimal weight of a face in planar triangulations without 4-vertices,” Mat. Zametki, vol. 51, no. 1, 16–19 (1992).
Borodin O. V., “Triangulated 3-polytopes with restricted minimal weight of faces,” Discrete Math., vol. 186, 281–285 (1998).
Avgustinovich S. V. and Borodin O. V., “Neighborhoods of edges in normal maps,” Diskretn. Anal. Issled. Oper., vol. 2, no. 2–3, 3–9 (1995).
Borodin O. V. and Ivanova A. O., “The vertex-face weight of edges in 3-polytopes,” Sib. Math. J., vol. 56, no. 2, 275–284 (2015).
Borodin O. V. and Ivanova A. O., “Heights of minor faces in triangle-free 3-polytopes,” Sib. Math. J., vol. 56, no. 5, 982–988 (2015).
Borodin O. V. and Loparev D. V., “The height of minor faces in normal plane maps,” Diskretn. Anal. Issled. Oper., Ser. 1, vol. 5, no. 4, 6–17 (1998).
Hornák M. and Jendrol’ S., “Unavoidable sets of face types for planar maps,” Discuss. Math. Graph Theory, vol. 16, no. 2, 123–142 (1996).
Jendrol’ S. and Voss H.-J., “Light subgraphs of graphs embedded in the plane—a survey,” Discrete Math., vol. 313, no. 4, 406–421 (2013).
Borodin O. V. and Woodall D. R., “The weight of faces in plane maps,” Mat. Zametki, vol. 64, no. 5, 648–657 (1998).
Borodin O. V. and Woodall D. R., “Cyclic degrees of 3-polytopes,” Graphs Comb., vol. 15, 267–277 (1999).
Borodin O. V., “Joint generalization of the theorems of Lebesgue and Kotzig on the combinatorics of planar maps,” Diskret. Mat., vol. 3, no. 4, 24–27 (1991).
Ferencová B. and Madaras T., “Light graphs in families of polyhedral graphs with prescribed minimum degree, face size, edge and dual edge weight,” Discrete Math., vol. 310, 1661–1675 (2010).
Jendrol’ S., “Triangles with restricted degrees of their boundary vertices in plane triangulations,” Discrete Math., vol. 196, 177–196 (1999).
Kotzig A., “Contribution to the theory of Eulerian polyhedra,” Mat. Eas. SAV (Math. Slovaca), vol. 5, 101–113 (1955).
Mohar B., Skrekovski R., and Voss H.-J., “Light subgraphs in planar graphs of minimum degree 4 and edge-degree 9,” J. Graph Theory, vol. 44, 261–295 (2003).
Wernicke P., “Über den kartographischen Vierfarbensatz,” Math. Ann., vol. 58, 413–426 (1904).
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The first author was supported by the Russian Foundation for Basic Research (Grants 15–01–05867 and 16–01–00499) 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. 58, No. 1, pp. 48–55, January–February, 2017; DOI: 10.17377/smzh.2017.58.105.
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Borodin, O.V., Ivanova, A.O. The height of faces of 3-polytopes. Sib Math J 58, 37–42 (2017). https://doi.org/10.1134/S0037446617010050
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DOI: https://doi.org/10.1134/S0037446617010050