Geodesic
Joint generalization of the theorems of Lebesgue and Kotzig on the combinatorics of planar maps
Diskretnaya Matematika, Tome 3 (1991) no. 4, pp. 24-27 Cet article a éte moissonné depuis la source Math-Net.Ru

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The weight of an edge in a map or polyhedron is the sum of the degrees of its end points. A map is normal if it does not contain vertices or faces incident to fewer than three edges. We prove that every planar normal map contains the following: either a 3-face incident to an edge of weight no greater than 13; or a 4-face incident to an edge of weight no greater than 8; or a 5-face incident to an edge of weight 6. All the bounds – 13, 8 and 6 – are attainable. 
@article{DM_1991_3_4_a3,
     author = {O. V. Borodin},
     title = {Joint generalization of the theorems of {Lebesgue} and {Kotzig} on the combinatorics of planar maps},
     journal = {Diskretnaya Matematika},
     pages = {24--27},
     year = {1991},
     volume = {3},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_1991_3_4_a3/}
}
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O. V. Borodin. Joint generalization of the theorems of Lebesgue and Kotzig on the combinatorics of planar maps. Diskretnaya Matematika, Tome 3 (1991) no. 4, pp. 24-27. http://geodesic.mathdoc.fr/item/DM_1991_3_4_a3/