Vertex-regular 11-factorizations in infinite graphs

The existence of $1$-factorizations of an infinite complete equipartite graph $K_m[n]$ (with $m$ parts of size $n$) admitting a vertex-regular automorphism group $G$ is known only when $n=1$ and $m$ is countable (that is, for countable complete graphs) and, in addition, $G$ is a finitely generated abelian group $G$ of order $m$. In this paper, we show that a vertex-regular $1$-factorization of $K_m[n]$ under the group $G$ exists if and only if $G$ has a subgroup $H$ of order $n$ whose index in $G$ is $m$. Furthermore, we provide a sufficient condition for an infinite Cayley graph to have a regular $1$-factorization. Finally, we construct 1-factorizations that contain a given subfactorization, both having a vertex-regular automorphism group

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

In this paper we consider decompositions of the complete graph Kv 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)

Abelian 1-factorizations of the complete graph

Extending a result by A. Hartman and A. Rosa [European J. Combin. 6 (1985), no. 1, 45--48], we prove that for any abelian group $G$ of even order, except for $G\simeq Z_{2^n}$ with $n>2$, there exists a one-factorization of the complete graph admitting $G$ as a sharply-vertex-transitive automorphism group

Nilpotent 1-factorizations of the complete graph

For which groups G of even order 2n does a 1-factorization of the complete graph on 2n veritces exist with the property of admitting G as a sharply vertex-transitive automorphism group? The complete answer is still unknown. Using the definition of a starter in G introduced in [M. Buratti "Abelian 1-factorizations of the complete graph" Europ. J Comb. 2001, pp.291-295], we give a positive answer for new classes of groups; for example, the nilpotent groups with either an abelian Sylow 2-subgroup or a non-abelian Sylow 2-subgroup which possesses a cyclic subgroup of index 2. Further considerations are given in case the automorphism group G fixes a 1-factor