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Theory on Structure and Coloring of Maximal Planar Graphs (2) Domino Configurations and Extending-Contracting Operations |
XU Jin |
(Key Laboratory of High Confidence Software Technologies, Peking University, Beijing 100871, China)
(School of Electronics Engineering and Computer Science, Peking University, Beijing 100871, China) |
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Abstract The first paper of this series of articles revealed that Four-Color Conjecture is hopefully proved mathematically by investigating a special class of graphs, called the 4-chromatic-funnel, pseudo uniquely-4- colorable maximal planar graphs. To characterize the properties of such class of graphs, a novel technique, “extending-contracting operation”, is proposed which can be used to construct maximal planar graphs. The essence of this technique is to study a special kind of configurations, domino configurations. In this paper, a necessary and sufficient condition for a planar graph to be a domino configuration is constructively given, on the basis of which it is proposed to construct the ancestor-graphs and descendent-graphs of a graph. Particularly, it is proved that every maximal planar graph with order n(≥9) and minimum degree ≥4 has an ancestor-graph of order (n-2) or (n-3) . Moreover, an approach is put forward to construct maximal planar graphs recursively, by which all maximal planar graphs with order 6~12 and minimum degree ≥4 are constructed. The extending-contracting operation constitutes the foundation in this series of articles.
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Received: 24 January 2016
Published: 26 April 2016
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Fund: The National 973 Program of China (2013CB 329600), The National Natural Science Foundation of China (61372191, 61472012, 61472433, 61572046, 61502012, 61572492, 61572153, 61402437) |
Corresponding Authors:
XU Jin
E-mail: jxu@pku.edu.cn
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