Approximating the permanent of graphs with large factors
Theoretical Computer Science
Bipolar orientations revisited
Discrete Applied Mathematics - Special issue: Fifth Franco-Japanese Days, Kyoto, October 1992
Embedding planar graphs on the grid
SODA '90 Proceedings of the first annual ACM-SIAM symposium on Discrete algorithms
On topological aspects of orientations
Discrete Mathematics
A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Dissections and trees, with applications to optimal mesh encoding and to random sampling
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Optimal coding and sampling of triangulations
ICALP'03 Proceedings of the 30th international conference on Automata, languages and programming
Convex drawings of 3-connected plane graphs
GD'04 Proceedings of the 12th international conference on Graph Drawing
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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.