12,281 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 bandwidth theorem for approximate decompositions
We provide a degree condition on a regular -vertex graph which ensures
the existence of a near optimal packing of any family of bounded
degree -vertex -chromatic separable graphs into . In general, this
degree condition is best possible.
Here a graph is separable if it has a sublinear separator whose removal
results in a set of components of sublinear size. Equivalently, the
separability condition can be replaced by that of having small bandwidth. Thus
our result can be viewed as a version of the bandwidth theorem of B\"ottcher,
Schacht and Taraz in the setting of approximate decompositions.
More precisely, let be the infimum over all
ensuring an approximate -decomposition of any sufficiently large regular
-vertex graph of degree at least . Now suppose that is an
-vertex graph which is close to -regular for some and suppose that is a sequence of bounded
degree -vertex -chromatic separable graphs with . We show that there is an edge-disjoint packing of
into .
If the are bipartite, then is sufficient. In
particular, this yields an approximate version of the tree packing conjecture
in the setting of regular host graphs of high degree. Similarly, our result
implies approximate versions of the Oberwolfach problem, the Alspach problem
and the existence of resolvable designs in the setting of regular host graphs
of high degree.Comment: Final version, to appear in the Proceedings of the London
Mathematical Societ
On oriented cycles in randomly perturbed digraphs
In 2003, Bohman, Frieze, and Martin initiated the study of randomly perturbed
graphs and digraphs. For digraphs, they showed that for every , there
exists a constant such that for every -vertex digraph of minimum
semi-degree at least , if one adds random edges then
asymptotically almost surely the resulting digraph contains a consistently
oriented Hamilton cycle. We generalize their result, showing that the
hypothesis of this theorem actually asymptotically almost surely ensures the
existence of every orientation of a cycle of every possible length,
simultaneously. Moreover, we prove that we can relax the minimum semi-degree
condition to a minimum total degree condition when considering orientations of
a cycle that do not contain a large number of vertices of indegree . Our
proofs make use of a variant of an absorbing method of Montgomery.Comment: 24 pages, 7 figures. Author accepted manuscript, to appear in
Combinatorics, Probability and Computin
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