1,152 research outputs found
Resolution of the Oberwolfach problem
The Oberwolfach problem, posed by Ringel in 1967, asks for a decomposition of
into edge-disjoint copies of a given -factor. We show that this
can be achieved for all large . We actually prove a significantly more
general result, which allows for decompositions into more general types of
factors. In particular, this also resolves the Hamilton-Waterloo problem for
large .Comment: 28 page
Infinitely many cyclic solutions to the Hamilton-Waterloo problem with odd length cycles
It is conjectured that for every pair of odd integers greater than
2 with , there exists a cyclic two-factorization of
having exactly factors of type and all the
others of type . The authors prove the conjecture in the affirmative
when and .Comment: 31 page
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
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
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