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An extension of Kotzig’s theorem. (English) Zbl 1350.05026
Summary: A. Kotzig [“Contribution to the theory of Eulerian polyhedra”, Mat.-Fyz. Čas., Slovensk. Akad. Vied 5, 101–113 (1955)] 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 $$d(u)\leq i$$ and $$d(v)\leq j$$. O. V. Borodin [Diskretn. Mat. 3, No. 4, 24–27 (1991; Zbl 0742.05034)] proved that every normal plane map contains an edge of one of the types (3, 10), (4, 7), or (5, 6), which is tight. R. Cole et al. [SIAM J. Discrete Math. 21, No. 1, 93–106 (2007; Zbl 1138.05059)] deduced from this result by Borodin that Kotzig’s bound of 13 is valid for all planar graphs with minimum degree $$\delta$$ at least 2 in which every $$d$$-vertex, $$d\geq 12$$, has at most $$d-11$$ neighbors of degree 2.
We give a common extension of the three above results by proving for any integer $$t\geq1$$ that every plane graph with $$\delta\geq2$$ and no $$d$$-vertex, $$d\geq11+t$$, 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.

##### MSC:
 05C15 Coloring of graphs and hypergraphs
##### Keywords:
plane graph; normal plane map; structural property; weight
Full Text:
##### References:
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