On the existence spectrum for sharply transitive -designs, a -matching

In this paper we consider decompositions of the complete graph K v into matchings of uniform cardinality k . They can only exist when k is an admissible value, that is a divisor of v ( v − 1 ) / 2 with 1 ≤ k ≤ v / 2 . The decompositions are required to admit an automorphism group Γ acting sharply transitively on the set of vertices. Here Γ is assumed to be either non-cyclic abelian or dihedral and we obtain necessary conditions for the existence of the decomposition when k is an admissible value with 1 < k < v / 2 . Differently from the case where Γ is a cyclic group, these conditions do exclude existence in specific cases. On the other hand we produce several constructions for a wide range of admissible values, in particular for every admissible value of k when v is odd and Γ is an arbitrary group of odd order possessing a subgroup of order gcd ( k , v ) .