曲面上色临界图点数的上界

Upper Bound for the Number of Vertices of Color-Critical Graphs on Surfaces

摘要:

Dirac观察到：对每个固定的曲面S和每个固定的自然数k ≥ 8，曲面S上仅有有限多个k-色临界图。Mohar和Thomassen证明了：对于亏格g ≥ 2的曲面S，曲面S上的7-色临界图的点数少于138(g ‒ 1)。我们借助于Euler公式和Gallai所发展起来的研究色临界图的方法，改进了这个结果，给出了曲面S上的7-色临界图的个数是有限的一个比较简洁的证明。除此以外，我们还给出曲面S上的每一个k-色临界图(k ≥ 7)的点数上界的一个统一的表达式。 Dirac observed that, for each fixed surface and each natural number k ≥ 8, there are only finitely many k-color-critical graphs on S. Mohar and Thomassen proved that for a surface S of genus g ≥ 2, every 7-color-critical graph on S has less than 138(g ‒ 1) vertices. Using Euler formula and the critical-graphs methods of Gallai, we improve this result and give a simple proof that the number of 7-color-critical graphs is finite. We also give unified expression for an upper bound of vertices of k-color-critical graphs (k ≥ 7) on surfaces.

文章引用：李青青, 晁福刚, 路伟华, 任韩. 曲面上色临界图点数的上界[J]. 应用数学进展, 2014, 3(2): 49-53. http://dx.doi.org/10.12677/AAM.2014.32008

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