Random cubic planar graphs

We show that the number of labeled cubic planar graphs on n vertices with n even is asymptotically αn-7/2ρ-nn!, where ρ-1 ... 3.13259 and α are analytic constants. We show also that the chromatic number of a random cubic planar graph that is chosen uniformly at random among all the labeled cubic planar graphs on n vertices is three with probability tending to e-ρ4/4! ... 0.999568 and four with probability tending to 1 - e-ρ4/4! as n → ∞ with n even. The proof given combines generating function techniques with probabilistic arguments. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007

[1]  W. T. Tutte,et al.  A Census of Planar Triangulations , 1962, Canadian Journal of Mathematics.

[2]  W. T. Tutte.  A Census of Planar Maps , 1963, Canadian Journal of Mathematics.

[3]  C. Itzykson,et al.  Quantum field theory techniques in graphical enumeration , 1980 .

[4]  Timothy R. S. Walsh,et al.  Counting labelled three-connected and homeomorphically irreducible two-connected graphs , 1982, J. Comb. Theory, Ser. B.

[5]  T. C. Hu,et al.  Combinatorial algorithms , 1982 .

[6]  Edward A. Bender,et al.  A survey of the asymptotic behaviour of maps , 1986, J. Comb. Theory, Ser. B.

[7]  Philippe Flajolet,et al.  A Calculus for the Random Generation of Labelled Combinatorial Structures , 1994, Theor. Comput. Sci..

[8]  Reinhard Diestel,et al.  Graph Theory , 1997 .

[9]  P. Francesco.  Matrix model combinatorics: applications to folding and coloring , 1999, math-ph/9911002.

[10]  Edward A. Bender,et al.  Asymptotics for the Probability of Connectedness and the Distribution of Number of Components , 2000, Electron. J. Comb..

[11]  Philippe Flajolet,et al.  Random maps, coalescing saddles, singularity analysis, and Airy phenomena , 2001, Random Struct. Algorithms.

[12]  P. Francesco,et al.  Census of planar maps: From the one-matrix model solution to a combinatorial proof , 2002, cond-mat/0207682.

[13]  Edward A. Bender,et al.  The Number of Labeled 2-Connected Planar Graphs , 2002, Electron. J. Comb..

[14]  N. Wormald,et al.  Enumeration of Rooted Cubic Planar Maps , 2002 .

[15]  Nicolas Bonichon,et al.  Canonical Decomposition of Outerplanar Maps and Application to Enumeration, Coding, and Generation , 2003, WG.

[16]  Deryk Osthus,et al.  On random planar graphs, the number of planar graphs and their triangulations , 2003, J. Comb. Theory, Ser. B.

[17]  R. Arratia,et al.  Logarithmic Combinatorial Structures: A Probabilistic Approach , 2003 .

[18]  P. Francesco,et al.  Combinatorics of hard particles on planar graphs , 2002, cond-mat/0211168.

[19]  Manuel Bodirsky,et al.  Generating Labeled Planar Graphs Uniformly at Random , 2003, ICALP.

[20]  C. Kiefer.  Quantum Gravity , 2004 .

[21]  2D QUANTUM GRAVITY,MATRIX MODELS AND GRAPH COMBINATORICS , 2004, math-ph/0406013.

[22]  Colin McDiarmid,et al.  On the Number of Edges in Random Planar Graphs , 2004, Combinatorics, Probability and Computing.

[23]  Omer Giménez,et al.  Asymptotic enumeration and limit laws of planar graphs , 2005, math/0501269.

[24]  Colin McDiarmid,et al.  Random planar graphs with n nodes and a fixed number of edges , 2005, SODA '05.

[25]  Manuel Bodirsky,et al.  Sampling Unlabeled Biconnected Planar Graphs , 2005, ISAAC.

[26]  M. Bodirsky,et al.  On the number of series parallel and outerplanar graphs , 2005 .

[27]  Colin McDiarmid,et al.  Random planar graphs , 2005, J. Comb. Theory B.

[28]  Dudley Stark.  LOGARITHMIC COMBINATORIAL STRUCTURES: A PROBABILISTIC APPROACH (EMS Monographs in Mathematics) By R ICHARD A RRATIA , A. D. B ARBOUR and S IMON T AVARÉ : 363 pp., €69.00, ISBN 3-03719-000-0 (European Mathematical Society, 2003) , 2005 .

[29]  Manuel Bodirsky,et al.  Generating Outerplanar Graphs Uniformly at Random , 2006, Combinatorics, Probability and Computing.

[30]  C. Gröpl,et al.  A direct decomposition of 3-connected planar graphs , 2007 .

[31]  C. McDiarmid,et al.  RANDOM PLANAR GRAPHS WITH GIVEN AVERAGE DEGREE , 2007 .