Fast generation of regular graphs and construction of cages

Abstract

The construction of complete lists of regular graphs up to isomorphism is one of the oldest problems in constructive combinatorics. In this paper an efficient algorithm to generate regular graphs with given number of vertices and vertex degree is introduced. The method is based on orderly generation refined by criteria to avoid isomorphism checking and combined with a fast test for canonicity. The implementation allows to compute even large classes of graphs, like construction of the 4-regular graphs on 18 vertices and, for the first time, the 5-regular graphs on 16 vertices. Also in cases with given girth some remarkable results are obtained. For instance the 5-regular graphs with girth 5 and minimal number of vertices were generated in less than one hour. There exist exactly four (5,5)-cages.

A conjecture on the prevalence of cubic bridge graphs

“…From numerical experiments using GENREG software [6] and the cubhamg utility in the package nauty [5] on cubic graphs of various orders, we observe that bridge graphs constitute the majority of non-Hamiltonian graphs. Moreover, as the graph order N increases, so does the ratio of cubic bridge graphs over all cubic non-Hamiltonian graphs of the same order.…”

About primitive regular graphs with exponent 2

“…Результаты для 4-, 5-и 6-регулярных графов представлены в таблице. Для генерации регулярных графов использовалась программы GENREG [5] и DSR Generator [6]. Вычисления показывают, что верхняя оценка может быть улучшена.…”

Using Novelty-Biased GA to Sample Diversity in Graphs Satisfying Constraints

“…This is further compounded by the constraint of a set global clustering coefficient. We therefore limited our investigations to possible configurations, as given in [16], with preliminary experiments used to determine realisable clustering coefficients by testing whether it was possible to find at least one valid network. We found that when limiting CF1 to its regularity component, it was always possible to find networks achieving a set level of global clustering.…”