# Description of faces in 3-polytopes without vertices of degree from 4 to 9

### Abstract

In 1940, Lebesgue proved that every normal plane map contains a face for which the set of degrees of its vertices is majorized by one of the following sequences: (3, 6,$\infty$), (3, 7, 41), (3, 8, 23), (3, 9, 17), (3, 10, 14), (3, 11, 13), (4, 4, $\infty$), (4, 5, 19), (4, 6, 11), (4, 7, 9), (5, 5, 9), (5, 6, 7), (3, 3, 3, $\infty$), (3, 3, 4, 11), (3, 3, 5, 7), (3, 4, 4, 5), (3, 3, 3, 3, 5). In this note prove that every 3-polytope without vertices of degree from 4 to 9 contains a face for which the set of degrees of its vertices is majorized by one of the following sequences: (3, 3, $\infty$), (3, 10, 12), (3, 3, 3, $\infty$), (3, 3, 3, 3, 3), which is tight.

### References

*Mathematical notes of NEFU*, 23(3), pp. 46-54. Available at: http://mzsvfu.ru/index.php/mz/article/view/description-of-faces-in-3-polytopes-without-vertices-of-degree-from-4-to-9 (Accessed: 22September2020).

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