We pose and solve the existence of 2-factorizations of complete graphs and complete bipartite graphs that have the number of cycles per 2-factor varying, called pancomponented. Such 2-factorizations exist for all such graphs. The pancomponented problem requires a slight generalization of the methods used to solve pancyclic 2-factorization problem, by building 2-factors from cyclically generated 1-factors. These two solutions are offered as the basic approaches to constructing the two essential parameters of a 2-factorization: the size and the number of cycles in the 2-factors.
Bibliographical noteFunding Information:
Dalibor Froncek would like to thank the Department of Mathematics and Statistics of the University of Vermont where he was visiting while the early versions of this paper were written. Later versions were in part supported by the University of Minnesota Duluth Grant 177-1009. Brett Stevens would like to acknowledge the support of PIMS, MITACS, Simon Fraser University and an NSERC Grant.
- Cycle decomposition
- Oberwolfach problem