Advertisement

The Pfaffian property of graphs on the Möbius strip based on topological resolution

  • Yan WangEmail author
Original Paper
  • 11 Downloads

Abstract

Recently, the problem about Pfaffian property of graphs has attracted much attention in the matching theory. Its significance stems from the fact that the number of perfect matchings in a graph G can be computed in polynomial time if G is Pfaffian. The Möbius strip with unique surface structure strip has potentially significant chirality properties in molecular chemistry. In this paper, we consider the Pfaffian property of graphs on the Möbius strip based on topological resolution. A sufficient condition for Pfaffian graphs on the Möbius strip is obtained. As its application, we characterize Pfaffian quadrilateral lattices on the Möbius strip.

Keywords

Möbius strip Pfaffian property Drawing in the plane with crossings 

Mathematics Subject Classification

05C10 05C70 05C75 92D20 

Notes

Acknowledgements

This work is partially supported by NSF of China(No. 11671186), NSF of Fujian Province (2017J01404) and Science Foundation for the Education Department of Fujian Province (JZ160455). It is also supported by the Institute of Meteorological Big Data-Digital Fujian and Fujian Key Laboratory of Data Science and Statistics.

References

  1. 1.
    J.A. Bondy, U.S.R. Murty, Graph Theory with Applications (Macmillan, London, 1978)Google Scholar
  2. 2.
    E.J. Cockayne, Chessboard domination problems. Discrete Math. 86, 13–20 (1990)CrossRefGoogle Scholar
  3. 3.
    P.A. Firby, C.F. Gardiner, Surface Topology, 2nd edn. (Ellis Horwood Series in Mathematics and its Applications, New York, 1991)Google Scholar
  4. 4.
    I. Fischer, C.H.C. Little, A characterization of Pfaffian near bipartite graphs. J. Comb. Theory Ser. B 82, 175–222 (2001)CrossRefGoogle Scholar
  5. 5.
    M.E. Fisher, Statistical mecanics of dimers on a plane lattice. Phys. Rev. 124, 1664–1672 (1961)CrossRefGoogle Scholar
  6. 6.
    F. Harary, P.G. Mezey, The diet transform of lattice patterns, equivalence relations, and similarity measures. Mol. Eng. 6, 415–416 (1996)CrossRefGoogle Scholar
  7. 7.
    F. Harary, P.G. Mezey, Cell-shedding transformations, equivalence relations, and similarity measures for square-cell configurations. Int. J. Quantum Chem. 62, 353–361 (1997)CrossRefGoogle Scholar
  8. 8.
    P.W. Kasteleyn, The statistics of dimers on a lattice I: the number of dimer arrangements on a quadratic lattice. Physica 27, 1209–1225 (1961)CrossRefGoogle Scholar
  9. 9.
    P.W. Kasteleyn, Dimer statistics and phase transitions. J. Math. Phys. 4, 287–293 (1963)CrossRefGoogle Scholar
  10. 10.
    P.W. Kasteleyn, Graph theory and crystal physics, in Graph Theory and Theoretical Physics, ed. by F. Harary (Academic Press, London, 1967), pp. 43–110Google Scholar
  11. 11.
    C.H.C. Little, A characterization of convertible (0, 1)-matrices. J. Comb. Theory 18, 187–208 (1975)CrossRefGoogle Scholar
  12. 12.
    L. Lovász, M. Plummer, Matching Theory. Annals of Discrete Mathematics, vol. 29 (North-Holland, New York, 1986)Google Scholar
  13. 13.
    F.L. Lu, L.Z. Zhang, F.G. Lin, Enumeration of perfect matchings of a type of quadratic lattice on the torus. Electron. J. Comb. 17, \(\sharp \)R36 (2010)Google Scholar
  14. 14.
    W. McCuaig, Pólya’s permanent problem. Electron. J. Comb. 11, \(\sharp \)R79 (2004)Google Scholar
  15. 15.
    P.G. Mezey, Topological tools for the study of families of reaction mechanisms: the fundamental groups of potential surfaces in the universal molecule context, in Applications of Topological Methods in Molecular Chemistry, ed. by R. Chauvin, C. Lepetit, B. Silvi, E. Alikhani (Springer, New York, 2016), pp. 243–255Google Scholar
  16. 16.
    B. Mohar, C. Thomassen, Graphs on Surface (The Johns Hopkins University Press, Baltimore, 2001)Google Scholar
  17. 17.
    J.R. Munkres, Elements of Algebraic Topology (Addison-Wesley, Menlo Park, 1984)Google Scholar
  18. 18.
    Norine, S.: Drawing Pfaffian graphs, graph drawing, in 12th International Symposium, LNCS, vol. 3383 (2005), pp. 371–376Google Scholar
  19. 19.
    N. Robertson, P.D. Seymour, R. Thomas, Permanent, Pfaffian orientations and even directed circuits. Math. Ann. 150, 929–975 (1999)CrossRefGoogle Scholar
  20. 20.
    G. Tesler, Matchings in graphs on non-orientable surfaces. J. Comb. Theory Ser. B 78, 198–231 (2000)CrossRefGoogle Scholar
  21. 21.
    Thomas, R.: A survey of Pfaffian orientations of graphs, in International Congress of Mathematicians, vol. III, (European Mathematical Society, Zurich, 2006), pp. 963–984Google Scholar
  22. 22.
    O.N. Temkin, A.V. Zeigarnik, D.G. Bonchev, Application of graph theory to chemical kinetics. Part 2. Topological specificity of single-route reaction mechanisms. J. Chem. Inf. Model 35(4), 729–737 (1995)CrossRefGoogle Scholar
  23. 23.
    L.G. Valiant, The complexity of computing the permanent. Theor. Comput. Sci. 8, 189–201 (1979)CrossRefGoogle Scholar
  24. 24.
    P.D. Walker, P.G. Mezey, Representation of square-cell configurations in the complex plane: tools for the characterization of molecular monolayers and cross sections of molecular surfaces. Int. J. Quantum Chem. 43(3), 375–392 (1992)CrossRefGoogle Scholar
  25. 25.
    Y. Wang, Pfaffian polyominos on the Klein bottle. J. Math. Chem. 56(10), 3147–3160 (2018)CrossRefGoogle Scholar
  26. 26.
    W.G. Yan, Y.N. Yeh, F.J. Zhang, Dimer problem on the cylinder and torus. Physica A 387, 6069–6078 (2008)CrossRefGoogle Scholar
  27. 27.
    L.Z. Zhang, Y. Wang, F.L. Lu, Pfaffian graphs embedding on the torus. Sci. China Math. 56, 1957–1964 (2013)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsMinnan Normal UniversityZhangzhouPeople’s Republic of China

Personalised recommendations