Generating Random Regular Graphs

  • Authors:

  • J. H. Kim;V. H. Vu

  • Affiliations:

  • Microsoft Research Microsoft Corporation, 98052, Redmond, WA, USA and Department of Mathematics, Yonsei University, Seoul, 120-749, South Korea;UCSD, Department of Mathematics, 92093, La Jolla, CA, USA

  • Venue:
  • Combinatorica
  • Year:
  • 2006

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Abstract

Random regular graphs play a central role in combinatorics and theoretical computer science. In this paper, we analyze a simple algorithm introduced by Steger and Wormald [10] and prove that it produces an asymptotically uniform random regular graph in a polynomial time. Precisely, for fixed d and n with d = O(n1/3−ε), it is shown that the algorithm generates an asymptotically uniform random d-regular graph on n vertices in time O(nd2). This confirms a conjecture of Wormald. The key ingredient in the proof is a recently developed concentration inequality by the second author. The algorithm works for relatively large d in practical (quadratic) time and can be used to derive many properties of uniform random regular graphs.