Generating Random Unlabelled Graphs

Abstract

Unlabelled graphs on n vertices can be generated uniformly at random, without calculating the total numbers of such graphs, but by using asymptotic enumeration results. For large n, the process is very efficient, taking $O(n^2 )$ steps on average. In fact, for $n \geqq 50$, virtually all practical applications will require only the first step of the algorithm to be implemented. This step is almost as simple as, and has the appearance of, the random generation of labelled graphs, yet the net effect is the uniform generation of unlabelled graphs at random. By similar methods one can generate random unlabelled graphs with n vertices and m edges, in expected time $O(n^2 \sqrt m )$ for most values of m. Also, random unlabelled r-regular graphs can be generated in expected time $O(n)$ when $r \geqq 3$ is fixed, but this is only (so far) practicable for a few small values of r.