86 research outputs found
Optimal path and cycle decompositions of dense quasirandom graphs
Motivated by longstanding conjectures regarding decompositions of graphs into
paths and cycles, we prove the following optimal decomposition results for
random graphs. Let be constant and let . Let be
the number of odd degree vertices in . Then a.a.s. the following hold:
(i) can be decomposed into cycles and a
matching of size .
(ii) can be decomposed into
paths.
(iii) can be decomposed into linear forests.
Each of these bounds is best possible. We actually derive (i)--(iii) from
`quasirandom' versions of our results. In that context, we also determine the
edge chromatic number of a given dense quasirandom graph of even order. For all
these results, our main tool is a result on Hamilton decompositions of robust
expanders by K\"uhn and Osthus.Comment: Some typos from the first version have been correcte
The existence of designs via iterative absorption: hypergraph -designs for arbitrary
We solve the existence problem for -designs for arbitrary -uniform
hypergraphs~. This implies that given any -uniform hypergraph~, the
trivially necessary divisibility conditions are sufficient to guarantee a
decomposition of any sufficiently large complete -uniform hypergraph into
edge-disjoint copies of~, which answers a question asked e.g.~by Keevash.
The graph case was proved by Wilson in 1975 and forms one of the
cornerstones of design theory. The case when~ is complete corresponds to the
existence of block designs, a problem going back to the 19th century, which was
recently settled by Keevash. In particular, our argument provides a new proof
of the existence of block designs, based on iterative absorption (which employs
purely probabilistic and combinatorial methods).
Our main result concerns decompositions of hypergraphs whose clique
distribution fulfills certain regularity constraints. Our argument allows us to
employ a `regularity boosting' process which frequently enables us to satisfy
these constraints even if the clique distribution of the original hypergraph
does not satisfy them. This enables us to go significantly beyond the setting
of quasirandom hypergraphs considered by Keevash. In particular, we obtain a
resilience version and a decomposition result for hypergraphs of large minimum
degree.Comment: This version combines the two manuscripts `The existence of designs
via iterative absorption' (arXiv:1611.06827v1) and the subsequent `Hypergraph
F-designs for arbitrary F' (arXiv:1706.01800) into a single paper, which will
appear in the Memoirs of the AM
Euler tours in hypergraphs
We show that a quasirandom -uniform hypergraph has a tight Euler tour
subject to the necessary condition that divides all vertex degrees. The
case when is complete confirms a conjecture of Chung, Diaconis and Graham
from 1989 on the existence of universal cycles for the -subsets of an
-set.Comment: version accepted for publication in Combinatoric
Pseudorandom hypergraph matchings
A celebrated theorem of Pippenger states that any almost regular hypergraph
with small codegrees has an almost perfect matching. We show that one can find
such an almost perfect matching which is `pseudorandom', meaning that, for
instance, the matching contains as many edges from a given set of edges as
predicted by a heuristic argument.Comment: 14 page
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
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