Advertisement

Graphs and Combinatorics

, Volume 27, Issue 1, pp 87–107 | Cite as

Random Planar Graphs with Bounds on the Maximum and Minimum Degrees

  • Chris DowdenEmail author
Original Paper
  • 59 Downloads

Abstract

Let P n,d,D denote the graph taken uniformly at random from the set of all labelled planar graphs on {1, 2, . . . , n} with minimum degree at least d(n) and maximum degree at most D(n). We use counting arguments to investigate the probability that P n,d,D will contain given components and subgraphs, showing exactly when this is bounded away from 0 and 1 as n → ∞.

Keywords

Planar graphs Random graphs Bounded degrees Labelled graphs 

Mathematics Subject Classification (2000)

05C10 05C80 05C07 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Dowden, C.: The evolution of uniform random planar graphs. Electron. J. Combin. 17(1), # R7 (2010)Google Scholar
  2. 2.
    Dowden, C.: Uniform random planar graphs with degree constraints. D.Phil. thesis, University of Oxford (2008). Available at http://ora.ouls.ox.ac.uk
  3. 3.
    Dowden, C., Addario-Berry, L.: Subgraphs of 4-regular planar graphs (submitted)Google Scholar
  4. 4.
    Gerke, S., McDiarmid, C., Steger, A., Weißl, A.: Random planar graphs with n nodes and a fixed number of edges. In: Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 999–1007 (2005)Google Scholar
  5. 5.
    Giménez O., Noy M.: Asymptotic enumeration and limit laws of planar graphs. J. Am. Math. Soc. 22, 309–329 (2009)CrossRefGoogle Scholar
  6. 6.
    McDiarmid C., Steger A., Welsh D.: Random planar graphs. J. Comb. Theory Ser. B 93, 187–205 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    van Lint J.H., Wilson R.M.: A Course in Combinatorics. Cambridge University Press, Cambridge (1992)zbMATHGoogle Scholar
  8. 8.
    Whitney H.: 2-Isomorphic graphs. Am. J. Math. 55, 245–254 (1933)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer 2010

Authors and Affiliations

  1. 1.LIX, École PolytechniquePalaiseau CedexFrance

Personalised recommendations