Abstract: The degree d of a vertex or face in a graph G is the number of incident edges. A face f = v,..,vd in a plane or torus graph G is of type (k1, k2,.... kd) if d(vi) ≤ ki for each i. By δ we denote the minimum vertex-degree of G. In 1989, Borodin confirmed Kotzig's conjecture of 1963 that every plane graph with minimum degree δ equal to 5 has a (5, 5, 7)-face or a (5, 6, 6)-face, where all parameters are tight. It follows from the classical theorem of Lebesgue (1940) that every plane quadrangulation with δ ≥ 3 has a face of one of the types (3, 3, 3,1), (3, 3, 4, 11), (3, 3, 5, 7), (3, 4, 4, 5). Recently, we improved this description to the following one: (3, 3, 3,1), (3, 3, 4, 9), (3, 3, 5, 6), (3, 4, 4, 5), where all parameters except possibly 9 are best possible and 9 cannot go down below 8. In 1995, Avgustinovich and Borodin proved that every torus quadrangulation with δ ≥ 3 has a face of one of the following types: (3, 3, 3,1), (3, 3, 4, 10), (3, 3, 5, 7), (3, 3, 6, 6), (3, 4, 4, 6), (4, 4, 4, 4), where all parameters are best possible. The purpose of our note is to prove that every torus triangulation with δ ≥ 5 has a face of one of the types (5, 5, 8), (5, 6, 7), or (6, 6, 6), where all parameters are best possible.

Cite: Borodin O.V. , Ivanova A.O.
TIGHT DESCRIPTION OF FACES IN TORUS TRIANGULATIONS WITH MINIMUM DEGREE 5
Сибирские электронные математические известия (Siberian Electronic Mathematical Reports). 2021. V.18. N2. P.1475-1481. DOI: 10.33048/SEMI.2021.18.110 WOS Scopus OpenAlex

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Web of science:	WOS:000734395000030
Scopus:	2-s2.0-85124154954
OpenAlex:	W4205869657

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