Transitive large sets of disjoint decompositions and group sequencings Original Research Article

Let n⩾3 be an integer, and let k denote either n or n−1. A large set of disjoint decompositions of Kn∗(Kn) into cycles of length k (denoted by k-LSD), is a partition of the set of all cycles of length k in Kn∗(Kn) into disjoint decompositions of Kn∗(Kn) (i.e., any two decompositions have no k-cycle in common). Such a large set is transitive, if there exists a permutation group on the vertices of Kn∗(Kn), which acts transitively on the decompositions in the large set. In this paper, we study the connection between group sequencings and the existence of transitive k-LSDs. We show that if there exists a sequenceable group of order n−1, then there exists a transitive n-LSD of Kn∗. Corresponding results are derived for the undirected case. We study also the connection between R-sequencings of groups and the existence of a transitive (n−1)-LSD of Kn∗. We derive necessary and sufficient conditions for the existence of a transitive k-LSD whose corresponding decompositions admit a regular group of automorphisms.