Counting Plane Graphs: Cross-Graph Charging Schemes

We study cross-graph charging schemes for graphs drawn in the plane. These are charging schemes where charge is moved across vertices of different graphs. Such methods have been recently applied to obtain various properties of triangulations that are embedded over a fixed set of points in the plane. We show how this method can be generalized to obtain results for various other types of graphs that are embedded in the plane. Specifically, we obtain a new bound of $O^*\left(187.53^N \right)$ for the maximum number of crossing-free straight-edge graphs that can be embedded over any specific set of N points in the plane (improving upon the previous best upper bound 207.85N in Hoffmann et al.[14]). We also derive upper bounds for numbers of several other types of plane graphs (such as connected and bi-connected plane graphs), and obtain various bounds on expected vertex-degrees in graphs that are uniformly chosen from the set of all crossing-free straight-edge graphs that can be embedded over a specific point set. We then show how to apply the cross-graph charging-scheme method for graphs that allow certain types of crossings. Specifically, we consider graphs with no set of k pairwise-crossing edges (more commonly known as k-quasi-planar graphs). For k=3 and k=4, we prove that, for any set S of N points in the plane, the number of graphs that have a straight-edge k-quasi-planar embedding over S is only exponential in N.

[1]  Csaba D. Tóth,et al.  Counting Plane Graphs: Flippability and Its Applications , 2011, WADS.

[2]  Micha Sharir,et al.  On degrees in random triangulations of point sets , 2011, J. Comb. Theory, Ser. A.

[3]  Marc Noy,et al.  Lower bounds on the number of crossing-free subgraphs of KN , 2000, Comput. Geom..

[4]  Eugène Catalan.  Solution nouvelle de cette question : un polygone étant donné, de combien de manières peut-on le partager en triangles au moyen de diagonales? , 1839 .

[5]  Emo Welzl,et al.  Counting Plane Graphs with Exponential Speed-Up , 2011, Rainbow of Computer Science.

[6]  Micha Sharir,et al.  Random triangulations of planar point sets , 2006, SCG '06.

[7]  Pavel Valtr.  Graph Drawings with no k Pairwise Crossing Edges , 1997, Graph Drawing.

[8]  Christian Sohler,et al.  Encoding a triangulation as a permutation of its point set , 1997, CCCG.

[9]  Ferran Hurtado,et al.  On the Number of Plane Geometric Graphs , 2007, Graphs Comb..

[10]  Csaba D. Tóth,et al.  Bounds on the maximum multiplicity of some common geometric graphs , 2011, STACS.

[11]  Heinrich Heesch,et al.  Untersuchungen zum Vierfarbenproblem , 1969 .

[12]  Raimund Seidel,et al.  A better upper bound on the number of triangulations of a planar point set , 2003, J. Comb. Theory, Ser. A.

[13]  Kevin Buchin,et al.  On the Number of Spanning Trees a Planar Graph Can Have , 2009, ESA.

[14]  Eyal Ackerman.  On the Maximum Number of Edges in Topological Graphs with no Four Pairwise Crossing Edges , 2009, Discret. Comput. Geom..

[15]  E. Szemerédi,et al.  Crossing-Free Subgraphs , 1982 .

[16]  Raimund Seidel,et al.  On the Number of Cycles in Planar Graphs , 2007, COCOON.

[17]  K. Appel,et al.  Every planar map is four colorable. Part I: Discharging , 1977 .

[18]  Jack Snoeyink,et al.  Number of Crossing-Free Geometric Graphs vs. Triangulations , 2008, Electron. Notes Discret. Math..

[19]  Philippe Flajolet,et al.  Analytic combinatorics of non-crossing configurations , 1999, Discret. Math..

[20]  Micha Sharir,et al.  Counting Triangulations of Planar Point Sets , 2009, Electron. J. Comb..

[21]  Gábor Tardos,et al.  On the maximum number of edges in quasi-planar graphs , 2007, J. Comb. Theory, Ser. A.

[22]  K. Appel,et al.  Every Planar Map Is Four Colorable , 2019, Mathematical Solitaires & Games.

[23]  János Pach.  Geometric Graph Theory , 2004, Handbook of Discrete and Computational Geometry, 2nd Ed..