On the existence spectrum for sharply transitive GG-designs, GG a [k][k]-matching

In this paper we consider decompositions of the complete graph KvKv into matchings of uniform cardinality kk. They can only exist when kk is an admissible value, that is a divisor of v(v−1)/2v(v−1)/2 with 1≤k≤v/21≤k≤v/2. The decompositions are required to admit an automorphism group ΓΓ acting sharply transitively on the set of vertices. Here ΓΓ is assumed to be either non-cyclic abelian or dihedral and we obtain necessary conditions for the existence of the decomposition when kk is an admissible value with 1