2,162 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
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
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