1 - Graph decompositions and symmetry

Summary

To the memory of Adriano Barlotti and Lucia Gionfriddo

Abstract

In this paper I shall try to review some results which were obtained in the area of factorizations and decompositions of complete graphs admitting an automorphism group with some specified properties. These properties primarily involve the action of the group on the objects of the decomposition, most often vertices, but also edges, subgraphs of the decomposition or factors of the factorization.

Classification theorems were obtained in highly symmetric situations, for example when the group acts doubly transitively on vertices, and it is often the case that all examples arise from geometry in this context.

A “less” symmetric situation involves a group acting sharply transitively on vertices, which means for any two given vertices there exists precisely one group element mapping the first vertex to the second one. The vertices of the complete graph can be identified with group elements in this case, and the decomposition or factorization can be described entirely within the group by techniques which are generally known as “difference” or “starter-like” methods. Existence may be a non-trivial question and generally depends on the isomorphism type of the chosen group.