Fast generation of cubic graphs

In this paper an efficient algorithm to generate regular graphs with small vertex valency is presented. The running times of a program based on this algorithm and designed to generate cubic graphs are below two natural benchmarks: (a) If N ( n ) denotes the number of pairwise non-isomorphic cubic graphs with n vertices and T ( n ) the time needed for generating the list of all these graphs, then T(n) /N(n ) decreases gradually for the observed values of n. This suggests that T(n) /N(n ) might be uniformly bounded for all n, ignoring the time to write the outputs, but we are unable to prove this and in fact are not confident about it. (b) For programs that generate lists of non-isomorphic objects, but cannot a priori make sure to avoid the generation of isomorphic copies, the time needed to check a randomly ordered list of these objects for being non-isomorphic is a natural benchmark. Since for large lists (n 2 22, girth 3) existing graph isomorphism programs take longer to canonically label all of the N ( n ) graphs than our algorithm takes to generate them, our algorithm is probably faster than any method which does one or more isomorphism test for every graph.