3,508 research outputs found
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
The Hamilton-Waterloo Problem with even cycle lengths
The Hamilton-Waterloo Problem HWP asks for a
2-factorization of the complete graph or , the complete graph with
the edges of a 1-factor removed, into -factors and
-factors, where . In the case that and are both
even, the problem has been solved except possibly when
or when and are both odd, in which case necessarily . In this paper, we develop a new construction that creates
factorizations with larger cycles from existing factorizations under certain
conditions. This construction enables us to show that there is a solution to
HWP for odd and whenever the obvious
necessary conditions hold, except possibly if ; and
; ; or . This result almost completely
settles the existence problem for even cycles, other than the possible
exceptions noted above
A Generalization of the Hamilton-Waterloo Problem on Complete Equipartite Graphs
The Hamilton-Waterloo problem asks for which and the complete graph
can be decomposed into copies of a given 2-factor and
copies of a given 2-factor (and one copy of a 1-factor if is even).
In this paper we generalize the problem to complete equipartite graphs
and show that can be decomposed into copies of a
2-factor consisting of cycles of length ; and copies of a 2-factor
consisting of cycles of length , whenever is odd, ,
and . We also give some more general
constructions where the cycles in a given two factor may have different
lengths. We use these constructions to find solutions to the Hamilton-Waterloo
problem for complete graphs
Quadratic forms and linear algebraic groups
Topics discussed at the workshop Quadratic Forms and Linear Algebraic Groups included besides the algebraic theory of quadratic and Hermitian forms and their Witt groups several aspects of the theory of linear algebraic groups and homogeneous varieties, as well as some arithmetic aspects pertaining to the theory of quadratic forms over function fields over local fields
- …