7 research outputs found
Vertex-regular -factorizations in infinite graphs
The existence of -factorizations of an infinite complete equipartite graph
(with parts of size ) admitting a vertex-regular automorphism
group is known only when and is countable (that is, for countable
complete graphs) and, in addition, is a finitely generated abelian group
of order .
In this paper, we show that a vertex-regular -factorization of
under the group exists if and only if has a subgroup of order
whose index in is . Furthermore, we provide a sufficient condition for
an infinite Cayley graph to have a regular -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)
Quaternionic 1-Factorizations and Complete Sets of Rainbow Spanning Trees
A 1-factorization F of a complete graph K2n
is said to be G-regular, or regular under G, if G is an automorphism group of F
acting sharply transitively on the vertex-set. The problem of determining which groups can realize such a situation dates back to a result by Hartman and Rosa (Eur J Comb 6:45–48, 1985) on cyclic groups and it is still open when n is even, although several classes of groups were tested in the recent past. It has been recently proved, see Rinaldi (Australas J Comb 80(2):178–196, 2021) and Mazzuoccolo et al. (Discret Math 342(4):1006–1016, 2019), that a G-regular 1-factorization, together with a complete set of rainbow spanning trees, exists for each group G of order 2n, n odd. The existence for each even n>2
was proved when either G is cyclic and n is not a power of 2, or when G is a dihedral group. Explicit constructions were given in all these cases. In this paper we extend this result and give explicit constructions when n>2
is even and G is either abelian but not cyclic, dicyclic, or a non cyclic 2-group with a cyclic subgroup of index 2
Sharply transitive 1-factorizations of complete multipartite graphs
Given a finite group G of even order, which graphs T have a 1-factorization admitting G as an automorphism group with a sharply transitive action on the vertex-set? Starting from this question we prove some general results and develop an exhustive analysis when T is a complete multipartite graph and G is cyclic
Sharply transitive 1-factorizations of complete multipartite graphs
Given a finite group G of even order, which graphs T have a 1-factorization admitting G as an automorphism group with a sharply transitive action on the vertex-set? Starting from this question we prove some general results and develop an exhustive analysis when T is a complete multipartite graph and G is cyclic
Frattini-based starters in 2-groups
Let G be a group of order 2^t , with t >3. We prove a sufficient condition for the existence of a one-factorization of a completegraph, admitting G as an automorphism group acting sharply transitively on the vertex-set