82 research outputs found

    Parallelism

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    EnProblems involving the idea of parallelism occur in finite geometry and in graph theory. This article addresses the question of constructing parallelisms with some degree of "symmetry". In particular, can we say anything on parallelisms admitting an automorphism group acting doubly transitively on "parallel classes"

    A complete solution to the infinite Oberwolfach problem

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    Let FF be a 22-regular graph of order vv. The Oberwolfach problem, OP(F)OP(F), asks for a 22-factorization of the complete graph on vv vertices in which each 22-factor is isomorphic to FF. In this paper, we give a complete solution to the Oberwolfach problem over infinite complete graphs, proving the existence of solutions that are regular under the action of a given involution free group GG. We will also consider the same problem in the more general contest of graphs FF that are spanning subgraphs of an infinite complete graph K\mathbb{K} and we provide a solution when FF is locally finite. Moreover, we characterize the infinite subgraphs LL of FF such that there exists a solution to OP(F)OP(F) containing a solution to OP(L)OP(L)

    On the full automorphism group of a Hamiltonian cycle system of odd order

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    It is shown that a necessary condition for an abstract group G to be the full automorphism group of a Hamiltonian cycle system is that G has odd order or it is either binary, or the affine linear group AGL(1; p) with p prime. We show that this condition is also sufficient except possibly for the class of non-solvable binary groups.Comment: 11 pages, 2 figure

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

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    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)

    A survey on constructive methods for the Oberwolfach problem and its variants

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    The generalized Oberwolfach problem asks for a decomposition of a graph GG into specified 2-regular spanning subgraphs F1,…,FkF_1,\ldots, F_k, called factors. The classic Oberwolfach problem corresponds to the case when all of the factors are pairwise isomorphic, and GG is the complete graph of odd order or the complete graph of even order with the edges of a 11-factor removed. When there are two possible factor types, it is called the Hamilton-Waterloo problem. In this paper we present a survey of constructive methods which have allowed recent progress in this area. Specifically, we consider blow-up type constructions, particularly as applied to the case when each factor consists of cycles of the same length. We consider the case when the factors are all bipartite (and hence consist of even cycles) and a method for using circulant graphs to find solutions. We also consider constructions which yield solutions with well-behaved automorphisms.Comment: To be published in the Fields Institute Communications book series. 23 pages, 2 figure

    Quaternionic 1-Factorizations and Complete Sets of Rainbow Spanning Trees

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    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&gt;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&gt;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
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