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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
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