|
|
Pure Mathematics 2020
C60的完美匹配与Hamilton圈
|
Abstract:
设G是一个3-正则的连通图。删掉G一个Hamilton圈(删边不删点)后剩下的子图是G的一个完美匹配；反之，删掉G一个完美匹配后剩下的子图只要是连通的，那一定是原图的Hamilton圈。因此本文通过删除完美匹配的方法给出了Buckminsterfullerene (C60)的所有Hamilton圈，进而通过Hamilton圈研究了完美匹配之间的关系。
Let G be a 3-regular connected graph. If we delete a Hamiltonian cycle of G (delete edges but not vertices), then the rest graph is a perfect matching of G. On the contrary, the rest subgraph deleting a perfect matching of G must be Hamiltonian cycle of G provided that it is connected. Consequently, all Hamiltonian cycles of C60 are given in this paper by the way of deleting perfect matchings. And then it is shown that the relationship of perfect matchings is obtained by Hamiltonian cycles.
| [1] | Kroto, H.W., Heath, J.R., O’Brien, S.C., Curl, R.F. and Smalley, R. (1985) C60: Buckminsterfullerene. Nature, 318, 162-163. https://doi.org/10.1038/318162a0 |
| [2] | Buckminsterfullerene, C60. http://www.chm.bris.ac.uk/motm/buckyball/c60a.htm |
| [3] | Klein, D.J., Schmalz, T.G., Hite, G.E., et al. (1986) Resonance in C60 Buckminsterfullerene. Journal of the American Chemical Society, 108, 1301-1302. https://doi.org/10.1021/ja00266a032 |
| [4] | Vuki？evi？, D., Kroto, H.W. and Randi？, M. (2005) Atlas of Kekulé Valence Structures of Buckminsterfullerene. Croatica Chemica Acta, 78, 223-234. |
| [5] | Vuki？evi？, D. and Randi？, M. (2011) Detailed Atlas of Kekulé Structures of the Buckminsterfullerene. The Mathematics and Topology of Fullerenes. Springer, Dordrecht, 153-169. https://doi.org/10.1007/978-94-007-0221-9_8 |
| [6] | Vuki？evi？, D. and Randi？, M. (2005) On Kekulé Structures of Buckmin-sterfullerene. Chemical Physics Letters, 401, 446-450. https://doi.org/10.1016/j.cplett.2004.11.098 |
| [7] | Flocke, N., Schmalz, T.G. and Klein, D.J. (1998) Variational Resonance Valence Bond Study on the Ground State of C60 Using the Heisenberg Model. Journal of Chemical Physics, 109, 873-880. https://doi.org/10.1063/1.476627 |
| [8] | Wu, J., Schmalz, T.G. and Klein. D.J. (2003) An Extended Heisenberg Model for conjugated Hydrocarbons. II. Kekulé Basis. The Journal of Chemical Physics, 119, 11011-11016. https://doi.org/10.1063/1.1622659 |
| [9] | Malkevitch, J. (1998) Polytopal Graphs. In: Beineke, L.W. and Wilson, R.J., Eds., Se-lected Topics in Graph Theory, Vol. 3, Academic Press, Amsterdam, 169-188. |
| [10] | Goodey, P.R. (1977) A Class of Hamiltonian Polytopes. Journal of Graph Theory, 1, 181-185. https://doi.org/10.1002/jgt.3190010213 |
| [11] | Aldred, R.E.L., Bau, S., Holton, D.A. and McKay, B.D. (2000) Nonhamiltonian 3-Connected Cubic Planar Graphs. SIAM Journal on Discrete Mathematics, 13, 25-32. https://doi.org/10.1137/S0895480198348665 |
| [12] | Kardo？, F. (2014) A Computer-Assisted Proof of Barnette-Goodey Conjecture: Not Only Fullerene Graphs Are Hamiltonian. SIAM Journal on Discrete Mathematics, 34, 62-100. https://doi.org/10.1137/140984737 |
| [13] | Bondy, J.A. and Murty, U.S.R. (1976) Graph Theory with Applications. The Macmillan Press Ltd., London and Basingstoke. https://doi.org/10.1007/978-1-349-03521-2 |
| [14] | 曾令辉. 富勒烯的完美匹配和反强迫数的计算[D]: [学士学位论文]. 兰州: 兰州大学, 2009. |
