Hostname: page-component-89b8bd64d-z2ts4 Total loading time: 0 Render date: 2026-05-12T04:33:34.691Z Has data issue: false hasContentIssue false
  • English
  • Français

Pegging Graphs Yields a Small Diameter

Published online by Cambridge University Press:  24 August 2010

STEFANIE GERKE ,
ANGELIKA STEGER  and
NICHOLAS WORMALD
STEFANIE GERKE
Affiliation:
Mathematics Department, Royal Holloway College, University of London, Egham, TW20 0EX, UK (e-mail: stefanie.gerke@rhul.ac.uk)
ANGELIKA STEGER
Affiliation:
Institute for Theoretical Computer Science, ETH Zurich, CH-8092 Zurich, Switzerland (e-mail: steger@inf.ethz.ch)
NICHOLAS WORMALD
Affiliation:
Department of Combinatorics and Optimization, University of Waterloo, Waterloo ON, N2L 3G1, Canada (e-mail: nwormald@uwaterloo.ca)
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the following process for generating large random cubic graphs. Starting with a given graph, repeatedly add edges that join the midpoints of two randomly chosen edges. We show that the growing graph asymptotically almost surely has logarithmic diameter. This process is motivated by a particular type of peer-to-peer network. Our method extends to similar processes that generate regular graphs of higher degree.

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 20 , Issue 2 , March 2011 , pp. 239 - 248
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1]Bollobás, B. and Chung, F. R. K. (1988) The diameter of a cycle plus a random matching. SIAM J. Discrete Math. 1 328–333.Google Scholar
[2]Bourassa, V. and Holt, F. (2003) SWAN: Small-world wide area networks. In Proc. International Conference on Advances in Infrastructures: SSGRR 2003w (L'Aquila, Italy, 2003), paper #64.Google Scholar
[3]Cooper, C., Dyer, M. and Greenhill, C. (2005) Sampling regular graphs and a peer-to-peer network. In Proc. Sixteenth Annual ACM–SIAM Symposium on Discrete Algorithms, pp. 980–988.Google Scholar
[4]Gao, P. Connectivity of random regular graphs generated by the pegging algorithm. J. Graph Theory, to appear.Google Scholar
[5]Gao, P. and Wormald, N. C. (2009) Short cycle distributions in random regular graphs recursively generated by pegging. Random Struct. Alg. 34 54–86.CrossRefGoogle Scholar
[6]Sanwalani, V. and Wormald, N. The diameter of random regular graphs. In preparation.Google Scholar
[7]Wormald, N. C. (1999) Models of random regular graphs. In Surveys in Combinatorics, 1999, Vol. 267 of London Mathematical Society Lecture Notes (Lamb, J. D. and Preece, D. A., eds), Cambridge University Press, Cambridge, pp. 239–298.Google Scholar
[8]Wormald, N. C. (1999) The differential equation method for random graph processes and greedy algorithms. In Lectures on Approximation and Randomized Algorithms (Karoñski, M. and Prömel, H. J., eds), PWN, Warsaw, pp. 73–155.Google Scholar
[9]Wormald, N. C. (2004) Random graphs and asymptotics. Section 8.2 in Handbook of Graph Theory (Gross, J. L. and Yellen, J., eds), CRC, Boca Raton, pp. 817–836.Google Scholar