论文信息 - On the Maximum Degree of a Random Planar Graph

On the Maximum Degree of a Random Planar Graph

Let the random graph Rn be drawn uniformly at random from the set of all simple planar graphs on n labelled vertices. We see that with high probability the maximum degree of Rn is Θ(ln n). We consider also the maximum size of a face and the maximum increase in the number of components on deleting a vertex. These results extend to graphs embeddable on any fixed surface.

Bruce A. Reed | Colin McDiarmid

[1] C. McDiarmid,et al. Random Graphs from Planar and Other Addable Classes , 2006 .

[2] H. Whitney. Congruent Graphs and the Connectivity of Graphs , 1932 .

[3] Nicholas C. Wormald,et al. The Distribution of the Maximum Vertex Degree in Random Planar Maps , 2000, J. Comb. Theory A.

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

[5] Colin McDiarmid,et al. Random graphs on surfaces , 2008, J. Comb. Theory, Ser. B.

[6] Eli Upfal,et al. On-line routing of random calls in networks , 2003 .

[7] Béla Bollobás,et al. Random Graphs , 1985 .

[8] Carsten Thomassen,et al. Graphs on Surfaces , 2001, Johns Hopkins series in the mathematical sciences.

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

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

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

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