极大平面图的结构与着色理论(2)多米诺构形与扩缩运算

许进

doi:10.11999/JEIT160224

极大平面图的结构与着色理论(2)多米诺构形与扩缩运算

doi: 10.11999/JEIT160224

许进¹

基金项目:

国家973规划项目(2013CB329600)，国家自然科学基金(61372191, 61472012, 61472433, 61572046, 61502012, 61572492, 61572153, 61402437)

计量
- 文章访问数: 1861
- HTML全文浏览量: 137
- PDF下载量: 874
- 被引次数: 0
出版历程
- 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: 业已证明四色猜想的数学证明可归结为刻画4-色漏斗型伪唯一4-色极大平面图的特征。为刻画此类极大平面图的结构特征，本文提出一种构造极大平面图的方法扩缩运算。研究发现：此方法的关键问题是需要清楚一种构形，称为多米诺构形。文中构造性地给出了多米诺构形的充要条件；在此基础上提出并建立了一个图的祖先图与子孙图理论与构造方法。特别证明了：任一最小度4的n(9)-阶极大平面图必含(n-2)-阶或(n-3)-阶祖先图；给出极大平面图的递推构造法，并用此方法构造出6~12-阶所有最小度的4极大平面图。扩缩运算是本系列文章的基石。
- 极大平面图 /
- 扩缩运算 /
- 多米诺构形 /
- 祖先图 /
- 子孙图 /
- 递推构造法
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

HTML全文

参考文献(26)

APPEL K and HAKEN W. The solution of the four-color map problem[J]. Science American, 1977, 237(4): 108-121. doi: 10.1038/scientificamerican1077-108.

APPEL K and HAKEN W. Every planar map is four colorable, I: Discharging[J]. Illinois Journal of Mathematics, 1977, 21(3): 429-490.

APPEL K, HAKEN W, and KOCH J. Every planar map is four-colorable, II: Reducibility[J]. Illinois Journal of Mathematics, 1977, 21(3): 491-567.

EBERHARD V. Zur Morphologie Der Polyeder, Mit Vielen Figuren Im Text[M]. Leipzig: Benedictus Gotthelf Teubner, 1891: 14-68.

王邵文. 构造极大平面图的圈加点法[J]. 北京机械工业学院学报, 2000, 15(1): 26-29.

WANG Shaowen. Method of cycle add-point to construct a maximum plate graph[J]. Journal of Beijing Institute of Machinery, 2000, 15(1): 26-29.

王邵文. 构造极大平面图的三种方法[J]. 北京机械工业学院学报, 1999, 14(1): 16-22.

WANG Shaowen. Three methods to construct maximum plain graph[J]. Journal of Beijing Institute of Machinery, 1999, 14(1): 16-22.

BARNETTE D. On generating planar graphs[J]. Discrete Mathematics, 1974, 7(3-4): 199-208. doi: 10.1016/0012- 365X(74)90035-1.

BUTLER J W. A generation procedure for the simple 3-polytopes with cyclically 5-connected graphs[J]. Journal of the Mechanical Behavior of Biomedical Materials, 1974, 26(2): 138-146.

BATAGELJ V. An inductive definition of the class of all triangulations with no vertex of degree smaller than 5[C]. Proceedings of the Fourth Yugoslav Seminar on Graph Theory, Novi Sad, 1983: 15-24.

WAGNER K. Bemerkungen zum vierfarbenproblem[J]. Jahresbericht der Deutschen Mathematiker-Vereinigung, 1936, 46: 26-32.

BRINKMANN G and MCKAY B D. Construction of planar triangulations with minimum degree 5[J]. Discrete Mathematics, 2005, 301: 147-163. doi: 10.1016/j.disc.2005.06. 019.

MCKAY B D. Isomorph-free exhaustive generation[J]. Journal of Algorithms, 1998, 26(2): 306-324. doi: 10.1006 /jagm.1997.0898.

AVIS D. Generating rooted triangulations without repetitions[J]. Algorithmica, 1996, 16(6): 618-632.

NAKANO S. Efficient generation of triconnected plane triangulations[J]. Computational Geometry, 2004, 27(2): 109-122.

BRINKMANN G and MCKAY B. Fast generation of planar graphs[J]. MATCH Communications in Mathematical and in Computer Chemistry, 2007, 58(58): 323-357.

NEGAMI S and NAKAMOTO A. Diagonal transformations of graphs on closed surfaces[J]. Science Reports of the Yokohama National University. Section I. Mathematics, Physics, Chemistry, 1994, 40(40): 71-96.

KOMURO H. The diagonal flips of triangulations on the sphere[J]. Yokohama Mathematical Journal, 1997, 44(2): 115-122.

MORI R, NAKAMOTO A, and OTA K. Diagonal flips in Hamiltonian triangulations on the sphere[J]. Graphs and Combinatorics, 2003, 19(3): 413-418. doi:?10.1007/s00373- 002-0508-6.

GAO Z C, URRUTIA J, and WANG J Y. Diagonal flips in labeled planar triangulations[J]. Graphs and Combinatorics, 2004, 17(4): 647-656. doi: ?10.1007/s003730170006.

BOSE P, JANSENS D, VAN RENSSEN A, et al. Making triangulations 4-connected using flips[C]. Proceedings of the 23rd Canadian Conference on Computational Geometry, Toronto, 2014, 47(2): 187-197 doi: 10.1016/j.comgeo.2012. 10.012.

许进. 极大平面图的结构与着色理论(1): 色多项式递推公式与四色猜想[J]. 电子与信息学报, 2016, 38(4): 763-779. doi: 10.11999/JEIT160072.

XU Jin. Theory on the structure and coloring of maximal planar graphs(1): recursion formula of chromatic polynomial and four-color conjecture[J]. Journal of Electronics Information Technology, 2016, 38(4): 763-779. doi: 10.11999/ JEIT160072.

BONDY J A and MURTY U S R. Graph Theory[M]. Springer, 2008: 5-46.

XU Jin, LI Zepeng, and ZHU Enqiang. On purely tree- colorable planar graphs[J]. Information Processing Letters, 2016, 116(8): 532-536. doi: 10.1016/j.ipl.2016.03.011.

施引文献

资源附件(0)

访问统计