# Low Faces of Restricted Degree in 3-Polytopes

- 3 Downloads

## Abstract

The degree of a vertex or face in a 3-polytope is the number of incident edges. A *k*-face is one of degree *k*, a *k*^{−}-face has degree at most *k*. The height of a face is the maximum degree of its incident vertices; 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; and so we assume the absence of pyramidal faces in what follows. In 1940, Lebesgue proved that each quadrangulated 3-polytope has a face *f* with *h*(*f*) ≤ 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 proved that *h* ≤ 20 for all triangulated 3-polytopes. In 1996, Horňák and Jendrol’ proved for arbitrarily polytopes that *h* ≤ 23. Recently, we obtained the sharp bounds *h* ≤ 10 for triangle-free polytopes and *h* ≤ 20 for arbitrary polytopes. Later, Borodin, Bykov, and Ivanova refined the latter result by proving that any polytope has a 10^{−}-face of height at most 20, where 10 and 20 are sharp. Also, we proved that any polytope has a 5^{−}-face of height at most 30, where 30 is sharp and improves the upper bound of 39 obtained by Horňák and Jendrol’ (1996). In this paper we prove that every polytope has a 6^{−}-face of height at most 22, where 6 and 22 are best possible. Since there is a construction in which every face of degree from 6 to 9 has height 22, we now know everything concerning the maximum heights of restricted-degree faces in 3-polytopes.

## Keywords

plane map planar graph 3-polytope structural properties height and degree of a face## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Steinitz E., “Polyeder und Raumeinteilungen,” Enzykl. Math. Wiss. (Geometrie), 3AB, vol. 12, 1–139 (1922).Google Scholar
- 2.Lebesgue H., “Quelques consequences simples de la formule d’Euler,” J. Math. Pures Appl., vol. 19, 27–43 (1940).MathSciNetzbMATHGoogle Scholar
- 3.Borodin O. V., “Colorings of plane graphs: a survey,” Discrete Math., vol. 313, no. 4, 517–539 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
- 4.Ore O. and Plummer M. D., “Cyclic coloration of plane graphs,” in:
*Recent Progress in Combinatorics*(W. T. Tutte, ed.), Acad. Press, New York, 1969, 287–293.Google Scholar - 5.Borodin O. V., “Strengthening Lebesgue’s theorem on the structure of the minor faces in convex polyhedra” (Russian), Diskretn. Anal. Issled. Oper., vol. 9, no. 3, 29–39 (2002).MathSciNetGoogle Scholar
- 6.Borodin O. V., “Solving the Kotzig and Grünbaum problems on the separability of a cycle in planar graphs” (Russian), Mat. Zametki, vol. 46, no. 5, 9–12 (1989); English translation: Math. Notes, vol. 46, no. 5–6, 835–837 (1989).zbMATHGoogle Scholar
- 7.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).MathSciNetCrossRefzbMATHGoogle Scholar
- 8.Borodin O. V., Ivanova A. O., and Kostochka A. V., “Describing faces in plane triangulations,” Discrete Math., vol. 319, no. 1, 47–61 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
- 9.Kotzig A., “From the theory of Eulerian polyhedra,” Mat. Cas., vol. 13, 20–31 (1963).zbMATHGoogle Scholar
- 10.Grunbaum B., “Polytopal graphs,” in:
*Studies in Graph Theory*(D. R. Fulkerson, ed.), MAA Stud. Math., vol. 12, 201–224 (1975).Google Scholar - 11.Plummer M. D., “On the cyclic connectivity of planar graph,” in:
*Graph Theory and Applications*, 303. Proc. Conf. Western Michigan Univ., Springer-Verlag, Berlin and Heidelberg, 1972, 235–242.CrossRefGoogle Scholar - 12.Kotzig A., “Extremal polyhedral graphs,” Ann. New York Acad. Sci., vol. 319, 569–570 (1979).Google Scholar
- 13.Borodin O. V., “Minimal weight of a face in planar triangulations without 4-vertices” (Russian), Mat. Zametki, vol. 51, no. 1, 16–19 (1992); English translation: Math. Notes, vol. 51, no. 1–2, 11–13 (1992).MathSciNetzbMATHGoogle Scholar
- 14.Borodin O. V., “Triangulated 3-polytopes with restricted minimal weight of faces,” Discrete Math., vol. 186, 281–285 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
- 15.Avgustinovich S. V. and Borodin O. V., “Neighborhoods of edges in normal maps” (Russian), Diskret. Anal. Issled. Oper., vol. 2, no. 3, 3–9 (1995).zbMATHGoogle Scholar
- 16.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).MathSciNetCrossRefzbMATHGoogle Scholar
- 17.Borodin O. V. and Ivanova A. O., “Heights of minor faces in triangle-free 3-polytopes,” Sib. Math. J., vol. 56, no. 5, 783–788 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
- 18.Borodin O. V. and Loparev D. V., “The height of minor faces in normal plane maps” (Russian), Diskret. Anal. Issled. Oper., vol. 5, no. 4, 6–17 (1998); English translation: Discrete Appl. Math., vol. 135, no. 1–3, 31–39 (2004).zbMATHGoogle Scholar
- 19.Horňák M. and Jendrol’ S., “Unavoidable sets of face types for planar maps,” Discuss. Math. Graph Theory, vol. 16, no. 2, 123–142 (1996).MathSciNetCrossRefzbMATHGoogle Scholar
- 20.Borodin O. V. and Ivanova A. O., “Low minor faces in 3-polytopes,” Discrete Math., vol. 341, no. 12, 3415–3424 (2018).MathSciNetCrossRefzbMATHGoogle Scholar
- 21.Borodin O. V. and Woodall D. R., “Cyclic degrees of 3-polytopes,” Graphs Combin., vol. 15, 267–277 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
- 22.Borodin O. V. and Woodall D. R., “The weight of faces in plane maps” (Russian), Mat. Zametki, vol. 6, no. 5, 648–657 (1998); English translation: Math. Notes, vol. 64, no. 5–6, 562–570 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
- 23.Ferencova B. and Madaras T., “Light graph in families of polyhedral graphs with prescribed minimum degree, face size, edge and dual edge weight,” Discrete Math., vol. 310, 1661–1675 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
- 24.Jendrol’ S., “Triangles with restricted degrees of their boundary vertices in plane triangulations,” Discrete Math., vol. 196, 177–196 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
- 25.Kotzig A., “Contribution to the theory of Eulerian polyhedra,” Mat.-Fyz. Casopis, vol. 5, 101–113 (1955).MathSciNetGoogle Scholar
- 26.Mohar B., Škrekovski 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).MathSciNetCrossRefzbMATHGoogle Scholar
- 27.Borodin O. V. and Ivanova A. O., “The height of faces of 3-polytopes,” Sib. Math. J., vol. 58, no. 1, 37–42 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
- 28.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).MathSciNetCrossRefzbMATHGoogle Scholar
- 29.Borodin O. V. and Ivanova A. O., “New results about the structure of plane graphs: A survey,” AIP Conf. Proc., vol. 1907, no. 1, 030051 (2017).CrossRefGoogle Scholar
- 30.Borodin O. V., Bykov M. A., and Ivanova A. O., “More about the height of faces in 3-polytopes,” Discuss. Math. Graph Theory, vol. 38, no. 2, 443–453 (2018).MathSciNetCrossRefzbMATHGoogle Scholar
- 31.Borodin O. V. and Ivanova A. O., “The weight of faces in normal plane maps,” Discrete Math., vol. 339, no. 10, 2573–2580 (2016).MathSciNetCrossRefzbMATHGoogle Scholar