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Height of minor faces in plane normal maps. (English. Russian original) Zbl 0913.05039
Discrete Appl. Math. 135, No. 1-3, 31-39 (2004); translation from Diskretn. Anal. Issled. Oper., Ser. 1 5, No. 4, 6-17 (1998).
The well-known Lebesgue theorem on planar normal maps (1940) and some more recent results by O. V. Borodin and S. V. Avgustinovich and by M. Horňák and S. Jendrol’ are strengthened. The main result is as follows: In any planar normal map without faces of the form \((4,4,\infty)\), \((3,5,\infty)\), \((3,3,3, \infty)\), there exists either a 3-face of height at most 20, or a 4-face of height at most 11, or a 5-face of height at most 5. Moreover, the estimates for 3- and 5-faces are unimprovable. The hypothesis is formulated that the above estimation for 4-faces is unimprovable, too.

MSC:
05C10 Planar graphs; geometric and topological aspects of graph theory
05C75 Structural characterization of families of graphs
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