Abstract
We deal with the asymptotic enumeration of combinatorial structures on planar maps. Prominent instances of such problems are the enumeration of spanning trees, bipartite perfect matchings, and ice models. The notion of an α-orientation unifies many different combinatorial structures, including the afore mentioned. We ask for the number of α-orientations and also for special instances thereof, such as Schnyder woods and bipolar orientations. The main focus of this paper are bounds for the maximum number of such structures that a planar map with n vertices can have. We give examples of triangulations with 2.37n Schnyder woods, 3-connected planar maps with 3.209n Schnyder woods and inner triangulations with 2.91n bipolar orientations. These lower bounds are accompanied by upper bounds of 3.56n, 8n, and 3.97n, respectively. We also show that for any planar map M and any α the number of α-orientations is bounded from above by 3.73n and we present a family of maps which have at least 2.598n α-orientations for n big enough.
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References
Baxter, R.J.: F model on a triangular lattice. J. Math. Physics 10, 1211–1216 (1969)
Bonichon, N.: A bijection between realizers of maximal plane graphs and pairs of non-crossing dyck paths. Discrete Mathematics, FPSAC 2002 Special Issue 298, 104–114 (2005)
Bonichon, N., Felsner, S., Mosbah, M.: Convex drawings of 3-connected planar graphs. In: Pach, J. (ed.) GD 2004. LNCS, vol. 3383, pp. 60–70. Springer, Heidelberg (2005)
Creed, P.: Counting Eulerian orientations is planar graphs is #P-complete. Personal Communication (2007)
Dagum, P., Luby, M.: Approximating the permanent of graphs with large factors. Theoretical Computer Science 102, 283–305 (1992)
de Fraysseix, H., de Mendez, P.O.: On topological aspects of orientation. Discrete Math. 229, 57–72 (2001)
de Fraysseix, H., de Mendez, P.O., Rosenstiehl, P.: Bipolar orientations revisited. Discrete Appl. Math. 56, 157–179 (1995)
Felsner, S.: Convex drawings of planar graphs and the order dimension of 3-polytopes. Order 18, 19–37 (2001)
Felsner, S.: Geometric Graphs and Arrangements. Vieweg Verlag (2004)
Felsner, S.: Lattice structures from planar graphs. Elec. J. Comb. R15 (2004)
Felsner, S., Huemer, C., Kappes, S., Orden, D.: Binary labelings for plane quadrangulations and their relatives (preprint, 2007)
Fusy, É., Poulalhon, D., Schaeffer, G.: Dissections and trees, with applications to optimal mesh encoding and to random sampling. In: SODA, pp. 690–699 (2005)
Jerrum, M., Sinclair, A., Vigoda, E.: A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries. In: ACM STOC, pp. 712–721 (2001)
Kenyon, R.W., Propp, J.G., Wilson, D.B.: Trees and matchings. Elec. J. Comb. 7 (2000)
Lieb, E.H.: The residual entropy of square ice. Physical Review 162, 162–172 (1967)
Lovász, L., Plummer, M.D.: Matching Theory. Annals of Discrete Mathematics, vol. 29. North-Holland, Amsterdam (1986)
Poulalhon, D., Schaeffer, G.: Optimal coding and sampling of triangulations. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 1080–1094. Springer, Heidelberg (2003)
Robertson, N., Seymour, P.D., Thomas, R.: Permanents, Pfaffian orientations, and even directed circuits. Ann. of Math. 150, 929–975 (1999)
Rote, G.: The number of spanning trees in a planar graph. In: Oberwolfach Reports. EMS, vol. 2, pp. 969–973 (2005), http://page.mi.fu-berlin.de/rote/about_me/publications.html
Schnyder, W.: Planar graphs and poset dimension. Order 5, 323–343 (1989)
Schnyder, W.: Embedding planar graphs on the grid. Proc. 1st ACM-SIAM Sympos. Discrete Algorithms 5, 138–148 (1990)
Tamassia, R., Tollis, I.G.: A unified approach to visibility representations of planar graphs. Discrete Comput. Geom. 1, 321–341 (1986)
Tutte, W.: A short proof of the factor theorem for finite graphs. Canadian Journal of Mathematics 6, 347–352 (1954)
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Felsner, S., Zickfeld, F. (2007). On the Number of α-Orientations. In: Brandstädt, A., Kratsch, D., Müller, H. (eds) Graph-Theoretic Concepts in Computer Science. WG 2007. Lecture Notes in Computer Science, vol 4769. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74839-7_19
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DOI: https://doi.org/10.1007/978-3-540-74839-7_19
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