Matrix analysis
A bound for the complexity of a simple graph
Discrete Mathematics
Chip-Firing and the Critical Group of a Graph
Journal of Algebraic Combinatorics: An International Journal
Introduction to Algorithms
Asymptotic Enumeration of Spanning Trees
Combinatorics, Probability and Computing
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Embedding 3-polytopes on a small grid
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
On the Number of Plane Geometric Graphs
Graphs and Combinatorics
On the number of cycles in planar graphs
COCOON'07 Proceedings of the 13th annual international conference on Computing and Combinatorics
Optimizing regular edge labelings
GD'10 Proceedings of the 18th international conference on Graph drawing
Counting plane graphs: flippability and its applications
WADS'11 Proceedings of the 12th international conference on Algorithms and data structures
Embedding stacked polytopes on a polynomial-size grid
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Realizing planar graphs as convex polytopes
GD'11 Proceedings of the 19th international conference on Graph Drawing
Counting plane graphs: cross-graph charging schemes
GD'12 Proceedings of the 20th international conference on Graph Drawing
On numbers of pseudo-triangulations
Computational Geometry: Theory and Applications
A simple aggregative algorithm for counting triangulations of planar point sets and related problems
Proceedings of the twenty-ninth annual symposium on Computational geometry
| Hi-index | 0.00 |
We prove that any planar graph on n vertices has less than O(5.2852n) spanning trees. Under the restriction that the planar graph is 3-connected and contains no triangle and no quadrilateral the number of its spanning trees is less than O(2.7156n). As a consequence of the latter the grid size needed to realize a 3d polytope with integer coordinates can be bounded by O(147.7n). Our observations imply improved upper bounds for related quantities: the number of cycle-free graphs in a planar graph is bounded by O(6.4884n), the number of plane spanning trees on a set of n points in the plane is bounded by O(158.6n), and the number of plane cycle-free graphs on a set of n points in the plane is bounded by O(194.7n).