Transitive large sets of disjoint decompositions and group sequencings (Q1613541)

Let \(K_n\) denote the complete graph of \(n\) vertices and \(K^*_n\) the complete directed graph on \(n\) vertices. Let \(n\geq 3\) be an integer, and let \(k\in \{n,n-1\}\). A large set of disjoint decompositions of \(K_n\) into cycles of length \(k\) (\(k\)-LSD) is a partition of the set of all cycles of length \(k\) in \(K_n\) into disjoint decompositions of \(K_n\). In the same way, a \(k\)-LSD for \(K^*_n\) is defined. A \(k\)-LSD is transitive, if there is a permutation group on the vertices of the graph, which acts transitively on the decompositions in the large set. The authors prove that there is a transitive \(n\)-LSD of \(K^*_n\), if there is a sequencable group of order \(n-1\). A similar result is obtained for the undirected case. Moreover, necessary and sufficient conditions for the existence of a transitive \(k\)-LSD whose corresponding decompositions admit a regular group of automorphisms are derived.