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.
plane map planar graph 3-polytope structural properties height and degree of a face
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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
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
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
Kotzig A., “From the theory of Eulerian polyhedra,” Mat. Cas., vol. 13, 20–31 (1963).zbMATHGoogle Scholar
Grunbaum B., “Polytopal graphs,” in: Studies in Graph Theory (D. R. Fulkerson, ed.), MAA Stud. Math., vol. 12, 201–224 (1975).Google Scholar
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
Kotzig A., “Extremal polyhedral graphs,” Ann. New York Acad. Sci., vol. 319, 569–570 (1979).Google Scholar
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
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
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
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