Spanning hypertrees, vertex tours and meanders (original) (raw)

Abstract

This paper revisits the notion of a spanning hypertree of a hypermap introduced by one of its authors and shows that it allows to shed new light on a very diverse set of recent results. The tour of a map along one of its spanning trees used by Bernardi may be generalized to hypermaps and we show that it is equivalent to a dual tour described by Cori [5] and Machì . We give a bijection between the spanning hypertrees of the reciprocal of the plane graph with 2 vertices and n parallel edges and the meanders of order n and a bijection of the same kind between semimeanders of order n and spanning hypertrees of the reciprocal of a plane graph with a single vertex and n/2 nested edges. We introduce hyperdeletions and hypercontractions in a hypermap which allow to count the spanning hypertrees of a hypermap recursively, and create a link with the computation of the Tutte polynomial of a graph. Having a particular interest in hypermaps which are reciprocals of maps, we generalize the reduction map introduced by Franz and Earnshaw to enumerate meanders to a reduction map that allows the enumeration of the spanning hypertrees of such hypermaps.

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References (29)

  1. O. Bernardi, A characterization of the Tutte polynomial via combinatorial embeddings, Ann. Comb. 12 (2008), 139-153.
  2. O. Bernardi, Tutte polynomial, subgraphs, orientations and sandpile model: new connections via embeddings, Electron. J. Combin. 15 (2008), Research Paper 109, 53 pp.
  3. O. Bernardi, T. Kalman and A. Postnikov, Universal Tutte polynomial, preprint 2020, arXiv:2004.00683 [math.CO].
  4. R. Cori, "Un code pour les Graphes Planaires et ses applications," Asterisque 27 (1975).
  5. R. Cori, Codage d'une carte planaire et hyperarbres recouvrants, Colloques Internat. C.N.R.S, Orsay (1976).
  6. R. Cori and A. Machì, Su alcune proprietà del genere di una coppia di permutazioni, Boll. Un. Mat. Ital. (5) 18-A (1981), 84-89.
  7. R. Cori and J-G. Penaud, The complexity of a planar hypermap and that of its dual, Ann. Discrete Math. 9 (1980), 53-62.
  8. P. Eadesand and C. F. X. de Mendonça Neto, Vertex splitting and tension-free layout, Graph drawing (Passau, 1995), 202-211, Lecture Notes in Comput. Sci., 1027, Springer, Berlin, 1996.
  9. P. Flajolet and R. Sedgewick, Analytic combinatorics, Cambridge University Press, Cambridge, 2009.
  10. P. Di Francesco, O. Golinelli and E. Guitter, Meander, folding, and arch statistics, Combinatorics and physics (Marseilles, 1995), Math. Comput. Modelling 26 (1997), 97-147.
  11. R.O.W. Franz, A partial order for the set of meanders, Ann. Comb. 2 (1998), 7-18.
  12. R.O.W. Franz, On the representation of meanders, Bayreuth. Math. Schr. (2003), 163-201.
  13. R.O.W. Franz and B.A. Earnshaw, A constructive enumeration of meanders, Ann. Comb. 6 (2002), 7-17.
  14. I. Goulden and A. Yong, Tree-like properties of cycle factorizations, J. Combin. Theory Ser. A 98 (2002), 106-117.
  15. A. Jacques, Sur le genre d'une paire de substitutions, C. R. Acad. Sci. Paris 267 (1968), 625-627.
  16. T. Kálmán, A version of Tutte's polynomial for hypergraphs, Adv. Math. 244 (2013), 823-873.
  17. T. Kálmán and A. Postnikov, Root polytopes, Tutte polynomials, and a duality theorem for bipartite graphs, Proc. Lond. Math. Soc. (3) 114 (2017), 561-588.
  18. T. Kálmán and L. Tóthmérész, Hypergraph polynomials and the Bernardi process, Algebr. Comb. 3 (2020), 1099-1139.
  19. G. Kreweras, Sur les partitions non croisées d'un cycle, Discrete Math. 1 (1972), 333-350.
  20. M. LaCroix, Approaches to the enumerative theory of meanders, Masters Essay, University of Wa- terloo, Waterloo, Ontario, Canada, 2003. https://www.math.uwaterloo.ca/ ~malacroi/Latex/ Meanders.pdf.
  21. S. Legendre, Foldings and meanders, Australas. J. Combin. 58 (2014), 275-291.
  22. W. F. Lunnon, A map-folding problem, Math. Comp. 22 (1968), 193-199.
  23. A. Machì, On the complexity of a hypermap, Discrete Math. 42 (1982), 221-226.
  24. N.J.A. Sloane, "On-Line Encyclopedia of Integer Sequences," http://www.research.att.com/\~njas/sequences
  25. P. Rosenstiehl, Planar permutations defined by two intersecting Jordan curves, Graph theory and combinatorics (Cambridge, 1983), 259-271, Academic Press, London, 1984.
  26. R. Simion and D. Ullman, On the structure of the lattice of noncrossing partitions, Discrete Math. 98 (1991), 193-206.
  27. J. Touchard, Contribution à l'étude du problème des timbres poste, Canad. J. Math. 2 (1950), 385-398.
  28. W. T. Tutte, Duality and trinity. Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vol. III, pp. 1459-472. Colloq. Math. Soc. János Bolyai, Vol. 10, North-Holland, Amsterdam, 1975.
  29. Labri, Université Bordeaux 1, 33405 Talence Cedex, France. WWW: http://www.labri.fr/perso/cori/. Department of Mathematics and Statistics, UNC Charlotte, Charlotte NC 28223- 0001. WWW: http://webpages.uncc.edu/ghetyei/.