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.
WANG Shaowen. Three methods to construct maximum plain graph[J]. Journal of Beijing Institute of Machinery, 1999, 14(1): 16-22.
[7]
BARNETTE D. On generating planar graphs[J]. Discrete Mathematics, 1974, 7(3-4): 199-208. doi: 10.1016/0012- 365X(74)90035-1.
[8]
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.
[9]
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.
[10]
WAGNER K. Bemerkungen zum vierfarbenproblem[J]. Jahresbericht der Deutschen Mathematiker-Vereinigung, 1936, 46: 26-32.
[11]
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.
[12]
MCKAY B D. Isomorph-free exhaustive generation[J]. Journal of Algorithms, 1998, 26(2): 306-324. doi: 10.1006 /jagm.1997.0898.
[13]
AVIS D. Generating rooted triangulations without repetitions[J]. Algorithmica, 1996, 16(6): 618-632.
[14]
NAKANO S. Efficient generation of triconnected plane triangulations[J]. Computational Geometry, 2004, 27(2): 109-122.
[15]
BRINKMANN G and MCKAY B. Fast generation of planar graphs[J]. MATCH Communications in Mathematical and in Computer Chemistry, 2007, 58(58): 323-357.
[16]
BRINKMANN G and MCKAY B D. The program plantri [OL]. http://cs.anu.edu.au/~bdm/plantri.
[17]
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.
[18]
KOMURO H. The diagonal flips of triangulations on the sphere[J]. Yokohama Mathematical Journal, 1997, 44(2): 115-122.
[19]
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.
[20]
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.
[21]
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.
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.
[23]
BONDY J A and MURTY U S R. Graph Theory[M]. Springer, 2008: 5-46.
[24]
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.
