648 research outputs found
Hurwitz equivalence of braid monodromies and extremal elliptic surfaces
We discuss the equivalence between the categories of certain ribbon graphs
and subgroups of the modular group and use it to construct
exponentially large families of not Hurwitz equivalent simple braid monodromy
factorizations of the same element. As an application, we also obtain
exponentially large families of {\it topologically} distinct algebraic objects
such as extremal elliptic surfaces, real trigonal curves, and real elliptic
surfaces
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
A complete solution to the infinite Oberwolfach problem
Let be a -regular graph of order . The Oberwolfach problem,
, asks for a -factorization of the complete graph on vertices in
which each -factor is isomorphic to . 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 . We will also consider the same problem in the more general
contest of graphs that are spanning subgraphs of an infinite complete graph
and we provide a solution when is locally finite. Moreover, we
characterize the infinite subgraphs of such that there exists a
solution to containing a solution to
On the full automorphism group of a Hamiltonian cycle system of odd order
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
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