Generating Random Regular Graphs Quickly

There are various algorithms known for generating graphs with n vertices of given degrees uniformly at random. Unfortunately, none of them is of practical use for all degree sequences, even for those with all degrees equal. In this paper we examine an algorithm which, although it does not generate uniformly at random, is provably close to a uniform generator when the degrees are relatively small. Moreover, it is easy to implement and quite fast in practice. The most interesting case is the regular one, when all degrees are equal to d = d(n) say. Moreover, methods for the regular case of this problem usually extend to arbitrary degree sequences, although the analysis can become more complicated and it may be needed to impose restrictions on the variation in the degrees (such as is analyzed by Jerrum et al. [4]). The first algorithm for generating d-regular graphs uniformly at random was implicit in the paper of Bollobás [2] and also in the approaches to counting regular graphs by Bender and Canfield [1] and in [13] (see also [14] for explicit algorithms). The configuration or pairing model of random d-regular graphs is as follows. Start with nd points (nd even) in n groups, and choose a random pairing of the points. Then create a graph with an edge from i to j if there is a pair containing points in the i’th and j’th groups. If no duplicated edge or loop (i.e., a pair of points in the same group) occurs,