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## Discussiones Mathematicae Graph Theory

2016 | 36 | 4 | 889-897
Tytuł artykułu

### An Extension of Kotzig’s Theorem

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In 1955, Kotzig proved that every 3-connected planar graph has an edge with the degree sum of its end vertices at most 13, which is tight. An edge uv is of type (i, j) if d(u) ≤ i and d(v) ≤ j. Borodin (1991) proved that every normal plane map contains an edge of one of the types (3, 10), (4, 7), or (5, 6), which is tight. Cole, Kowalik, and Škrekovski (2007) deduced from this result by Borodin that Kotzig’s bound of 13 is valid for all planar graphs with minimum degree δ at least 2 in which every d-vertex, d ≥ 12, has at most d − 11 neighbors of degree 2. We give a common extension of the three above results by proving for any integer t ≥ 1 that every plane graph with δ ≥ 2 and no d-vertex, d ≥ 11+t, having more than d − 11 neighbors of degree 2 has an edge of one of the following types: (2, 10+t), (3, 10), (4, 7), or (5, 6), where all parameters are tight.
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889-897
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wydano
2016-11-01
otrzymano
2015-10-09
poprawiono
2016-01-06
zaakceptowano
2016-01-06
online
2016-10-21
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• Institute of Mathematics Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090,, brdnoleg@math.nsc.ru
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