Generation of Cubic graphs

Cubic or 3-regular graphs are (simple) graphs where each vertex has degree 3. The class of cubic graphs is especially interesting for mathematical applications because for various important open problems in graph theory cubic graphs are the smallest or simplest possible potential counterexamples. In chemistry, cubic graphs serve as models for the Nobel Prize winning fullerenes [14] or, more generally, for some cyclopolyenes [2]. The generation of cubic graphs can be considered a benchmark problem in structure enumeration. The first complete lists of cubic connected graphs were given by de Vries at the end of the 19th century, who gave a list of all cubic (connected) graphs up to 10 vertices [9, 10]. The first computer approach was by Balaban, a theoretical chemist, in 1966/67 who generated all cubic graphs up to 12 vertices [2]. De Vries’ lists were independently confirmed by hand by Bussemaker and Seidel in 1968 [8] and Imrich in 1971 [13]. From 1974 on, various algorithms for the generation of cubic graphs were published. Each algorithm was implemented in a computer program that could generate larger lists of cubic graphs, see [21, 11, 7, 18, 3]. In 1983 Robinson and Wormald [23] published a paper on the non-constructive enumeration of cubic graphs. When the present research was begun, the fastest publicly available program for the generation of cubic graphs was minibaum [3]. When developed in 1992, minibaum could be used to generate complete lists of all cubic graphs up to 24 vertices and several restricted classes with more vertices, like bipartite graphs or graphs with higher girth. Later, when more and faster computers were available, minibaum was used to generate all cubic graphs up to 30 vertices in order to test them for Yutsis decompositions [1].