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.
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* Research supported in part by grant RB091G-VU from UCSD, by NSF grant DMS-0200357 and by an A. Sloan fellowship.
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Kim, J.H., Vu*, V.H. Generating Random Regular Graphs. Combinatorica 26, 683–708 (2006). https://doi.org/10.1007/s00493-006-0037-7
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DOI: https://doi.org/10.1007/s00493-006-0037-7