科研之友
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摘要
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. S krekovski (2007) 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 >= 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 delta >= 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.
- 出版日期2016-11
