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A Tutte Polynomial for Maps

  • ANDREW GOODALL (a1), THOMAS KRAJEWSKI (a2), GUUS REGTS (a3) and LLUÍS VENA (a1) (a4)
Abstract

We follow the example of Tutte in his construction of the dichromate of a graph (i.e. the Tutte polynomial) as a unification of the chromatic polynomial and the flow polynomial in order to construct a new polynomial invariant of maps (graphs embedded in orientable surfaces). We call this the surface Tutte polynomial. The surface Tutte polynomial of a map contains the Las Vergnas polynomial, the Bollobás–Riordan polynomial and the Krushkal polynomial as specializations. By construction, the surface Tutte polynomial includes among its evaluations the number of local tensions and local flows taking values in any given finite group. Other evaluations include the number of quasi-forests.

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[1] Askanazi, R., Chmutov, S., Estill, C., Michel, J. and Stollenwerk, P. (2013) Polynomial invariants of graphs on surfaces. Quantum Topol. 4 7790.
[2] Biggs, N. (1993) Algebraic Graph Theory, second edition, Cambridge Mathematical Library, Cambridge University Press.
[3] Bollobás, B. (1998) Modern Graph Theory, Vol. 184 of Graduate Texts in Mathematics, Springer.
[4] Bollobás, B. and Riordan, O. (2001) A polynomial invariant of graphs on orientable surfaces. Proc. London Math. Soc. (3) 83 513531.
[5] Bollobás, B. and Riordan, O. (2002) A polynomial of graphs on surfaces. Math. Ann. 323 8196.
[6] Bouchet, A. (1989) Maps and ▵-matroids. Discrete Math. 78 5971.
[7] Butler, C. (2018) A quasi-tree expansion of the Krushkal polynomial. Adv. in Appl. Math., 94 322.
[8] Champanerkar, A., Kofman, I. and Stoltzfus, N. (2011) Quasi-tree expansion for the Bollobás–Riordan–Tutte polynomial. Bull. Lond. Math. Soc. 43 972984.
[9] Chun, C., Moffatt, I., Noble, S. and Rueckeriemen, R. (2016) Matroids, delta-matroids and embedded graphs. arXiv:1403.0920
[10] DeVos, M. J. (2000) Flows on graphs. PhD thesis, Princeton University.
[11] Edmonds, J. K. (1960) A Combinatorial Representation for Polyhedral Surfaces, Notices of the American Mathematical Society, AMS.
[12] Ellis-Monaghan, J. A. and Merino, C. (2011) Graph polynomials and their applications I: The Tutte polynomial. In Structural Analysis of Complex Networks, Birkhäuser/Springer, pp. 219255.
[13] Ellis-Monaghan, J. A. and Moffatt, I. (2013) Graphs on Surfaces, Springer Briefs in Mathematics, Springer.
[14] Ellis-Monaghan, J. A. and Moffatt, I. (2015) The Las Vergnas polynomial for embedded graphs. European J. Combin. 50 97114.
[15] Frobenius, G. (1896) Über Gruppencharaktere. Sitzber. Königlich Preuss Akad. Wiss. Berlin, pp. 9851021.
[16] Goodall, A., Litjens, B., Regts, G. and Vena, L. (2017) A Tutte polynomial for non-orientable maps. Electron. Notes Discrete Math. 61 513519.
[17] Hatcher, A. (2002) Algebraic Topology, Cambridge University Press.
[18] Jones, G. A. (1998) Characters and surfaces: A survey. In The Atlas of Finite Groups: Ten Years On, Vol. 249 of London Mathematical Society Lecture Note Series, Cambridge University Press, pp. 90118.
[19] Kochol, M. (2002) Polynomials associated with nowhere-zero flows. J. Combin. Theory Ser. B 84 260269.
[20] Krushkal, V. (2011) Graphs, links, and duality on surfaces. Combin. Probab. Comput. 20 267287.
[21] Lando, S. K. and Zvonkin, A. K. (2004) Graphs on Surfaces and their Applications, Vol. 141 of Encyclopaedia of Mathematical Sciences, Springer.
[22] Las Vergnas, M. (1978) Sur les activités des orientations d'une géométrie combinatoire. Cahiers Centre Études Rech. Opér. 20 293300.
[23] Las Vergnas, M. (1980) On the Tutte polynomial of a morphism of matroids, Ann. Discrete Math. 8 720.
[24] Litjens, B. (2017) On dihedral flows in embedded graphs. arXiv:1709.06469
[25] Litjens, B. and Sevenster, B. Partition functions and a generalized coloring-flow duality for embedded graphs. J. Graph Theory. http://doi.org/10.1002/jgt.22210
[26] Mednyh, A. D. (1978) Determination of the number of nonequivalent coverings over a compact Riemann surface. Dokl. Akad. Nauk SSSR 239 269271.
[27] Mulase, M. and Yu, J. T. (2005) Non-commutative matrix integrals and representation varieties of surface groups in a finite group. Ann. Inst. Fourier (Grenoble) 55 21612196.
[28] Noble, S. D. and Welsh, D. J. A. (1999) A weighted graph polynomial from chromatic invariants of knots. Ann. Inst. Fourier (Grenoble) 49 10571087.
[29] Oxley, J. G. and Welsh, D. J. A. (1979) The Tutte polynomial and percolation. In Graph Theory and Related Topics (Bondy, J. A. et al., eds), Academic Press, pp. 329339.
[30] Serre, J.-P. (2012) Linear Representations of Finite Groups, Vol. 42 of Graduate Texts in Mathematics, Springer.
[31] Tutte, W. T. (1947) A ring in graph theory. Proc. Cambridge Philos. Soc. 43 2640.
[32] Tutte, W. T. (1949) On the imbedding of linear graphs in surfaces. Proc. London Math. Soc. (2) 51 474483.
[33] Tutte, W. T. (1954) A contribution to the theory of chromatic polynomials. Canad. J. Math. 6 8091.
[34] Tutte, W. T. (2001) Graph Theory, Vol. 21 of Encyclopedia of Mathematics and its Applications, Cambridge University Press.
[35] Tutte, W. T. (2004) Graph-polynomials. Adv. Appl. Math. 32 59.
[36] Watanabe, Y. and Fukumizu, K. (2011) New graph polynomials from the Bethe approximation of the Ising partition function. Combin. Probab. Comput. 20 299320.
[37] Welsh, D. J. A. (1993) Complexity: Knots, Colourings and Counting, Vol. 186 of London Mathematical Society Lecture Note Series, Cambridge University Press.
[38] Welsh, D. J. A. (1999) The Tutte polynomial. Random Struct. Alg. 15 210228.
[39] Whitney, H. (1932) Non-separable and planar graphs. Trans. Amer. Math. Soc. 34 339362.
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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
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