Z-transformation graphs of perfect matchings of plane bipartite graphs: A survey

Match

Volumn 56, Issue 3, 2006, Pages 457-476

  Zhang, Heping ^a  

^a LANZHOU UNIVERSITY   (China)

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EID: 33846504437     PISSN: 03406253     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Review

References (74)

1. S. Bau, M.A. Henning, Matching transformation graphs of cubic bipartite plane graphs, Discrete Math. 262 (2003) 27-36.
2. G. Birkhoff, Lattice Theory, Third edition, American Math. Soc., Providence, 1979.
3. Z. Che, Z. Chen, Forcing hexagons in hexagonal systems, MATCH Commun. Math. Comput. Chem., 2006, this issue.
4. R. Chen, F. Zhang, Hamilton paths in Z-transformation graphs of perfect matchings of hexagonal systems, Discrete Appl. Math. 74 (1997) 191-196.
5. Z. Chen, Directed tree structure of the set of Kekulé patterns of polyhex graphs. Chem. Phy. lett. 115(3) (1985) 291-293.
6. N.G. Chopra, R.J. Luyken, K. Cherrey, V.H. Crespi, M.L. Cohen, S.G. Louie, A. Zettl, Boron-nitride nanotubes, Science 269 (1995) 966-967.
7. S.J. Cyvin, I. Gutman, Kekulé Structures in Benzenoid Hydrocarbons, Springer, Berlin, 1988.
8. S.J. Cyvin, J. Brunvoll and B.N. Cyvin, Theory of Coronoid Hydrocarbons, Lectures Notes in Chemistry 54, Springer, Berlin, 1991.
9. S. El-Basil, Kekulé structures as graph generators, J. Math. Chem. 14 (1993) 305-318.
10. S. El-Basil, Generation of lattice graphs. An equivalence relation on Kekulé counts of catacondensed benzenoid hydrocarbons, J. Mol. Struct. (Theochem.) 288 (1993) 67-84.
11. S. Felsner, Lattice structures from planar graphs, Elect. J. Combin. 11 (2004), #R15.
12. J.C. Fournier, Combinatorics of perfect, matchings in plane bipartite graphs and application to tilings, Theor. Comput. Sci. 303 (2003) 333-351.
13. P.W. Fowler, T. Heine, D. Mitchell, R. Schmidt, G. Seifert, Boron-nitrogen analogues of the fullerenes: The isolated-square rule, J. Chem. Soc. Faraday Trans. 92 (12) (1996) 2197-2201.
14. G. Grätzer, Lattice Theory, First Concepts and Distributive Lattice, W. H. Freeman and Company, San Francisco, 1971.
15. W. Gründler, Signifikante Elektronenstrukturen fur benzenoide Kohlenwasserstoffe, Wiss. Z. Univ. Halle 31 (1982) 97-116.
16. W. Gründler, Mesomerie und Quantenmechanic, Z. Chem. (Lepizig) 23 (1983) 157-167.
17. I. Gutman, Topological properties of benzenoid systems, in: Advances in the Theory of Benzenoid Hydrocarbons II, Topics in Current Chem. 162, Springer, Berlin, 1992.
18. I. Gutman, S.J. Cyvin, Introduction to the Theory of Benzenoid Hydrocarbons, Springer-Verlag, Berlin, 1989.
19. I. Gutman, S. Klavzar, Chemical graph theory of Fibonacenes, MATCH Commun. Math. Comput. Chem. 55(1) (2006) 39-54.
20. P. Hansen, M. Zheng, Bonds fixed by fixing bonds, J. Chem. Inform. Comput. Sci. 34 (1994) 297-304.
21. P. Hansen and M. Zheng, Upper bounds for the Clar number of benzenoid hydrocarbons, J. Chem. Soc., Faraday Trans. 88 (1992) 1621-1625.
22. P. Hansen and M. Zheng, The Clar number of a benzenoid hydrocarbon and linear programming, J. Math. Chem. 15 (1994) 93-107.
23. F. Harary, D.J. Klein, T.P. Zivkovic, Graphical properties of polyhexes: perfect matching vector and forcing, J. Math. Chem. 6 (1991) 295-306.
24. W.C. Herndon, Resonance energies of aromatic hydrocarbons. Quantitative test of resonance theory, J. Am. Chem. Soc. 95 (7) (1973) 2404-2406.
25. W.-J. Hsu, Fibonacci cubes - A new interconnection topology, IEEE Trans. Parallel Distr. Systems 4 (1993) 3-12.
26. W.-J. Hsu, C.V. Page, J.-S. Liu, Fibonacci cubes - A class of self-similar graphs, Fibonacci Quart. 31 (1993) 65-72.
27. W. Imrich, S. Klavžar, Product Graphs: Structure and Recognition, John Wiley & Sons, New York, 2000.
28. S. Khuller, J. Naor, P. Klein, The lattice structure of flow in planar graphs, SIAM J. Discrete Math. 6(3) (1993) 477-490.
29. S. Klavžar, A. Vesel, P. Žigert. On resonance graphs of catacondensed hexagonal graphs: structure, coding, and Hamiltonian path algorithm, MATCH Commun. Math. Comput. Chem. 49 (2003) 99-116.
30. S. Klavžar, A. Vesel, P. Žigert, I. Gutman, Binary coding of Kekulé structures of catacondensed benzenoid hydrocarbons, Comput. Chem. 25 (2001) 569-575.
31. S. Klavžar, P. Žigert, Fibonacci cubes are the resonance graphs of fibonaccenes, Fibonacci Quart. 43 (3) (2005) 269-276.
32. S. Klavžar, P. Žigert, A min-max result on catacondensed benzenoid graphs, Appl. Math. Lett. 15 (2002) 279-283
33. S. Klavžar, P. Žigert, G. Brinkmann, Resonance graphs of catacondensed even ring systems are median, Discrete Math. 253 (2002) 35-43.
34. S. Klavžar, P. Žigert, I. Gutman, Clar number of catacondensed benzenoid hydrocarbons, J. Mol. Struct. (Theochem) 586 (2002) 235-240.
35. P.C.B. Lam, W.C. Shiu, H. Zhang, Elementary blocks of plane bipartite graphs, MATCH Commun. Math. Comput. Chem. 49 (2003) 127-137
36. P.C.B. Lam, W.C. Shiu, H. Zhang, Resonance graphs and a binary coding for the 1-factors of benzenoid systems, Preprint, 2004.
37. P.C.B. Lam, H. Zhang, A distributive lattice on the set of perfect matchings of a plane bipartite graph, Order 20 (2003) 13-29.
38. L. Lovász, M. D. Plummer, Matching Theory, Annals of Discrete Math. 29, North-Holland, Amsterdam, 1986.
39. O. Pretzel, On reorienting graphs by pushing down maximal vertices. II, Discrete Math. 270 (2003) 227-240.
40. J. Propp, Lattice structure for orientations of graphs, Preprint 1993, http://www.math.wisc.edu/~propp/orient/.
41. W. R. Pulleyblank, Polyhedral combinatorics, in: Mathematical Programming, The State of the Art, A. Bachem, M. Grotschel and B. Korte, Eds., Springer-Verlag, Berlin, 1983, pp.312-345.
42. M. Randić, Aromaticity of polycyclic conjugated hydrocarbons, Chem. Reviews 103 (9) (2003) 3449-3605.
43. M. Randić, Resonance in catacondensed benzenoid hydrocarbons, Int. J. Quantum Chem. 63(2) (1997) 585-600.
44. M. Randić, D.J. Klein, S. El-Basil, P. Calkins, Resonance in large benzenoid hydrocarbons, Croat. Chem. Acta 69 (4) (1996) 1639-1660.
45. K. Salem, S. Klavžar, I. Gutman, On the role of hypercubes in the resonance graphs of benzenoid graphs, Discrete Math. 306 (2006) 699-704.
46. T. Saldanha, R. Casarin, Spaces of domino tilings, Discrete Comput. Geom. 14 (1995) 207-233.
47. W.C. Shiu, P.C.B. Lam, F. Zhang, H. Zhang, Normal components, Kekulé patterns, and Clar patterns in plane bipartite graphs, J. Math. Chem. 31(4) (2002) 405-420.
48. P. R. Stanley, Enumerative Combinatorics, Volume 1, Wadsworth, Belmont, CA, 1986.
49. A. Vesel, Characterization of resonance graphs of catacondensed hexagonal graphs, MATCH Commun. Math. Comput. Chem. 53 (2005) 195-208.
50. D. Wang, The connectivity of a class of transformation graphs, J. Mah. Study 29(2) (1996) 76-80.
51. F. Zhang and R. Chen, When each hexagon of a hexagonal system covers it, Discrete Appl. Math. 30 (1991) 63-75.
52. F. Zhang, X. Guo, The classification of hexagonal systems by their Z-transformation graphs, J. Xinjiang Univ. Natur. Sci. 4 (1987) 7-12. (Chinese)
53. F. Zhang, X. Guo, The enumeration of several classes of hexagonal systems. Acta Math. Appl. Sinica (English Ser.) 15 (1999) 65-71.
54. F. Zhang, X. Guo, R. Chen, Z-transformation graph of perfect matchings of hexagonal systems, Discrete Math. 72 (1988) 405-415.
55. F. Zhang, X. Guo, R. Chen, The connectivity of Z-transformation graphs of perfect matchings of hexagonal systems. Acta Math. Appl. Sinica 4(2)(1988) 131-135.
56. F. Zhang, X. Li, Forcing bonds of a benzenoid system, Acta Math. Appl. Sinica., Engl. Ser. 12(2) (1996) 209-215.
57. F. Zhang, X. Li, Hexagonal systems with forcing edges, Discrete Math. 140 (1995) 253-263.
58. F. Zhang, X. Li, H. Zhang, Hexagonal systems with fixed bonds, Discrete Appl. Math. 47(3) (1993) 285-296.
59. F. Zhang, H. Zhang, A new enumeration method for Kekulé structures of hexagonal systems with forcing edges, J. Mol. Struct. (Theochem) 331 (1995) 255-260.
60. F. Zhang, Y. Liu, Estimation of the resonance energy of benzenoid hydrocarbon, Chinese Science Bulletin 38 (1993) 2040-2043.
61. F. Zhang, M. Zheng, Generalized hexagonal systems with each hexagon being resonant, Discrete Appl. Math. 36 (1992) 67-73.
62. H. Zhang, The connectivity of Z-transformation graphs of perfect matchings of polyominoes, Discrete Math. 158 (1996) 257-272.
63. H. Zhang, Direct sum of distributive lattices on the set of perfect matchings of a plane bipartite graph, Preprint.
64. H. Zhang, D. Yang, H. Yao, Decomposition theorem on matchable distributive lattices, preprint, 2005.
65. H. Zhang, H. Yao, D. Yang, A min-max result on outerplane bipartite graphs, Appl. Math. Lett., in press.
66. H. Zhang, R. Zha, H. Yao, Z-transformation graphs of maximum matchings of plane bipartite graphs, Discrete Appl. Math., 134 (2004) 339-350.
67. H. Zhang, F. Zhang, The rotation graphs of perfect matchings of plane bipartite graphs, Discrete Appl. Math. 73 (1997) 5-12.
68. H. Zhang, F. Zhang, Block graphs of Z-transformation graphs of perfect matchings of plane elementary bipartite graphs, Ars Combin. 53 (1999) 309-314.
69. H. Zhang, F. Zhang, Total Z-transformation graph of perfect matchings of plane bipartite graph, Electronic Notes in Discrete Math. Vol. 5(2000).
70. H. Zhang, F. Zhang, Plane elementary bipartite graphs, Discrete Appl. Math. 105 (2000) 291-311.
71. H. Zhang, F. Zhang, Perfect matchings of polyomino graphs, Graphs Combin. 13 (1997) 295-304.
72. H. Zhang, F. Zhang, H. Yao, Z-transformation graphs of perfect matchings of plane bipartite graphs, Discrete Math. 276 (2004) 393-404.
73. H. Zhang, L. Zhao, H. Yao, The Z-transformation graph for an outerplane bipartite graph has a Hamilton path, Appl. Math. Lett. 17/8 (2004) 897-901.
74. L. Zhang, Z-transformation graph of perfect matchings of catacondensed hexagonal system, J. Mah. Study 31(4) (1998) 437-441.