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Resonance Graphs on Perfect Matchings of Graphs on Surfaces

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Abstract

Let G be a graph embedded in a surface and let be the set of all faces of G. For a given subset \(\mathcal F\) of even faces (faces bounded by an even cycle), the resonance graph of G with respect to \({\mathcal {F}}\), denoted by \(R(G;\mathcal F)\), is a graph such that its vertex set is the set of all perfect matchings of G and two vertices \(M_1\) and \(M_2\) are adjacent if and only if the symmetric difference \(M_1\oplus M_2\) is a cycle bounding some face in \({\mathcal {F}}\). It has been shown that if G is a plane elementary bipartite graph, the resonance graph of G with respect to the set of all inner faces is isomorphic to the covering graph of a distributive lattice. However, this result does not hold in general for plane graphs G. The structure properties of resonance graphs in general remain unknown. In this paper, we show that every connected component of the resonance graph of a graph G on a surface with respect to a given even-face set can always be embedded into a hypercube as an induced subgraph. Further, we show that the Clar covering polynomial of G with respect to \({\mathcal {F}}\) is equal to the cube polynomial of the resonance graph \(R(G;{\mathcal {F}})\).

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References

  1. Agnarsson, G.: Induced subgraphs of hypercubes. Eur. J. Combin. 34, 155–168 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Avann, S.P.: Metric ternary distributive semi-lattices. Proc. Am. Math. Soc. 12, 407–414 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bandelt, H.-J.: Retracts of hypercubes. J. Gr. Theory 8, 501–510 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  4. Berlič, M., Tratnik, N., Žigert Pleteršek, P.: Equivalence of Zhang-Zhang polynomial and cube polynomial for spherical benzenoid systems. MATCH Commun. Math. Comput. Chem. 73, 443–456 (2015)

    MathSciNet  MATH  Google Scholar 

  5. Brešar, B., Imrich, W., Klavžar, S., Mulder, H.M., Škrekovski, R.: Tiled partial cubes. J. Gr. Theory 40, 91–103 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brešar, B., Klavžar, S., Škrekovski, R.: The cube polynomial and its derivatives: the case of median graphs. Electron. J. Combin. 10, #R3 (2003)

  7. Brešar, B., Klavžar, S., Lipovec, A., Mohar, B.: Cubic inflation, mirror graphs, regular maps and partial cubes. Eur. J. Combin. 25, 55–64 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brešar, B., Klavžar, S., Škrekovski, R.: Roots of cube polynomials of median graphs. J. Gr. Theory 52, 37–50 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Che, Z.: Structural properties of resonance graphs of plane elementary bipartite graphs. Discrete Appl. Math. 247, 102–110 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  10. Duffus, D., Rival, I.: Graphs orientable as distributive lattices. Proc. Am. Math. Soc. 88, 197–200 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  11. Edmonds, J.: Maximum matching and a polyhedron with (0, 1) vertices. J. Res. Nat. Bur. Stand. B 69, 125–130 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  12. Felsner, S.: Lattice structures from planar graphs. Electron. J. Combin. 11, #R15 (2004)

  13. Felsner, S., Knauer, K.: Distributive lattices, polyhedra, and generalized flows. Eur. J. Combin. 32, 45–59 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Garey, M.R., Graham, R.L.: On cubical graphs. J. Combin. Theory Ser. B 18, 84–95 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gründler, W.: Signifikante Elektronenstrukturen fur benzenoide Kohlenwasserstoffe. Wiss. Z. Univ. Halle 31, 97–116 (1982)

    Google Scholar 

  16. Hammack, R., Imrich, W., Klavžar, S.: Handbook of Product Graphs, 2nc edn. CRC Press, Boca Raton (2011)

    Book  MATH  Google Scholar 

  17. Klavžar, S., Vesel, A., Žigert, P., Gutman, I.: Binary coding of Kekulé structures of catacondensed benzenoid hydrocarbons. Comput. Chem. 25, 569–575 (2001)

    Article  Google Scholar 

  18. Klavžar, S., Žigert, P., Brinkmann, G.: Resonance graphs of catacondensed even ring systems are median. Discrete Math. 253, 35–43 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lam, P.C.B., Zhang, H.: A distributive lattice on the set of perfect matchings of a plane bipartite graph. Order 20, 13–29 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lovász, L.: Matching structure and the matching lattice. J. Combin. Theory Ser. B 43, 187–222 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lovász, L., Plummer, M.D.: Matching Theory. North-Holland, Amsterdam (1986)

    MATH  Google Scholar 

  22. Ovchinnikov, S.: Graphs and Cubes, pp. 89–125. Springer, New York (2011)

    Book  MATH  Google Scholar 

  23. Plummer, M.D.: On \(n\)-extendable graphs. Discrete Math. 31, 201–210 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  24. Propp, J.: Lattice structure for orientations of graphs. arXiv:math/0209005

  25. Salem, K., Klavžar, S., Vesel, A., Žigert, P.: The Clar formulas of a benzenoid system and the resonance graph. Discrete Appl. Math. 157, 2565–2569 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Tratnik, N., Žigert Pleteršek, P.: Distributive lattice structure on the set of perfect matchings of carbon nanotubes. J. Math. Chem. 54, 1296–1305 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. Tratnik, N., Žigert Pleteršek, P.: Resonance graphs of fullerenes. Ars Math. Contemp. 11, 425–435 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zhang, H.: \(Z\)-transformation graphs of perfect matchings of plane bipartite graphs: a survey. MATCH Commun. Math. Comput. Chem. 56, 457–476 (2006)

    MathSciNet  MATH  Google Scholar 

  29. Zhang, H., Zhang, F.: The Clar covering polynomial of hexagonal systems I. Discrete Appl. Math. 69, 147–167 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zhang, H., Zhang, F.: Plane elementary bipartite graphs. Discrete Appl. Math. 105, 291–311 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  31. Zhang, F., Guo, X., Chen, R.: \(Z\)-transformation graphs of perfect matchings of hexagonal systems. Discrete Math. 72, 405–415 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zhang, H., Zhang, F., Yao, H.: \(Z\)-transformation graphs of perfect matchings of plane bipartite graphs. Discrete Math. 276, 393–404 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  33. Zhang, H., Lam, P.C.B., Shiu, W.C.: Resonance graphs and a binary coding for the 1-factors of benzenoid systems. SIAM J. Discrete Math. 22, 971–984 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zhang, H., Ou, L., Yao, H.: Fibonacci-like cubes as \(Z\)-transformation graphs. Discrete Math. 309, 1284–1293 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  35. Zhang, H., Shiu, W.C., Sun, P.K.: A relation between Clar covering polynomial and cube polynomial. MATCH Commun. Math. Comput. Chem. 70, 477–492 (2013)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments to improve the final version of the paper.

Funding

This article was funded by Slovenian Research Agency (Grant no. P1-0297 and the bilateral project between Slovenia and the USA - project No. BI-US/22-24-158) and Simons Foundation (Grant no. 359516).

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Authors and Affiliations

  1. Faculty of Natural Sciences and Mathematics, University of Maribor, Maribor, Slovenia

    Niko Tratnik

  2. Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia

    Niko Tratnik

  3. Department of Mathematical Sciences and Center for Computational and Data Sciences, Middle Tennessee State University, Murfreesboro, TN, 37132, USA

    Dong Ye

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  1. Niko Tratnik
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Niko Tratnik supported by the Slovenian Research Agency (research core funding no. P1-0297) and the bilateral project between Slovenia and the USA (Project no. BI-US/22-24-158). Dong Ye partially supported by a grant from the Simons Foundation (no. 359516).

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Tratnik, N., Ye, D. Resonance Graphs on Perfect Matchings of Graphs on Surfaces. Graphs and Combinatorics 39, 68 (2023). https://doi.org/10.1007/s00373-023-02666-4

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  • DOI: https://doi.org/10.1007/s00373-023-02666-4

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