極大平面圖的結構與著色理論(2)多米諾構形與擴縮運算

許進

doi:10.11999/JEIT160224

極大平面圖的結構與著色理論(2)多米諾構形與擴縮運算

doi: 10.11999/JEIT160224

許進¹

基金項目:

國家973規(guī)劃項目(2013CB329600)，國家自然科學基金(61372191, 61472012, 61472433, 61572046, 61502012, 61572492, 61572153, 61402437)

計量
- 文章訪問數(shù): 1913
- HTML全文瀏覽量: 148
- PDF下載量: 874
- 被引次數(shù): 0
出版歷程
- 收稿日期: 2016-01-24
- 修回日期: 2016-04-21
- 刊出日期: 2016-06-19

Theory on Structure and Coloring of Maximal Planar Graphs (2) Domino Configurations and Extending-Contracting Operations

XU Jin¹

Funds:

The National 973 Program of China (2013CB 329600), The National Natural Science Foundation of China (61372191, 61472012, 61472433, 61572046, 61502012, 61572492, 61572153, 61402437)

摘要

摘要: 業(yè)已證明四色猜想的數(shù)學證明可歸結為刻畫4-色漏斗型偽唯一4-色極大平面圖的特征。為刻畫此類極大平面圖的結構特征，本文提出一種構造極大平面圖的方法擴縮運算。研究發(fā)現(xiàn)：此方法的關鍵問題是需要清楚一種構形，稱為多米諾構形。文中構造性地給出了多米諾構形的充要條件；在此基礎上提出并建立了一個圖的祖先圖與子孫圖理論與構造方法。特別證明了：任一最小度4的n(9)-階極大平面圖必含(n-2)-階或(n-3)-階祖先圖；給出極大平面圖的遞推構造法，并用此方法構造出6~12-階所有最小度的4極大平面圖。擴縮運算是本系列文章的基石。
- 極大平面圖 /
- 擴縮運算 /
- 多米諾構形 /
- 祖先圖 /
- 子孫圖 /
- 遞推構造法
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 ordern(9) and minimum degree4 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.
- Maximal planar graphs /
- Extending-contracting operations /
- Domino configurations /
- Ancestor-graphs /
- Descendent-graphs /
- Recursive construction approach

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許進. 極大平面圖的結構與著色理論(1): 色多項式遞推公式與四色猜想[J]. 電子與信息學報, 2016, 38(4): 763-779. doi: 10.11999/JEIT160072.

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