218 research outputs found
Spanning trees of 3-uniform hypergraphs
Masbaum and Vaintrob's "Pfaffian matrix tree theorem" implies that counting
spanning trees of a 3-uniform hypergraph (abbreviated to 3-graph) can be done
in polynomial time for a class of "3-Pfaffian" 3-graphs, comparable to and
related to the class of Pfaffian graphs. We prove a complexity result for
recognizing a 3-Pfaffian 3-graph and describe two large classes of 3-Pfaffian
3-graphs -- one of these is given by a forbidden subgraph characterization
analogous to Little's for bipartite Pfaffian graphs, and the other consists of
a class of partial Steiner triple systems for which the property of being
3-Pfaffian can be reduced to the property of an associated graph being
Pfaffian. We exhibit an infinite set of partial Steiner triple systems that are
not 3-Pfaffian, none of which can be reduced to any other by deletion or
contraction of triples.
We also find some necessary or sufficient conditions for the existence of a
spanning tree of a 3-graph (much more succinct than can be obtained by the
currently fastest polynomial-time algorithm of Gabow and Stallmann for finding
a spanning tree) and a superexponential lower bound on the number of spanning
trees of a Steiner triple system.Comment: 34 pages, 9 figure
Hypergraph matchings and designs
We survey some aspects of the perfect matching problem in hypergraphs, with
particular emphasis on structural characterisation of the existence problem in
dense hypergraphs and the existence of designs.Comment: 19 pages, for the 2018 IC
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
Finding an almost perfect matching in a hypergraph avoiding forbidden submatchings
In 1973, Erd\H{o}s conjectured the existence of high girth -Steiner
systems. Recently, Glock, K\"{u}hn, Lo, and Osthus and independently Bohman and
Warnke proved the approximate version of Erd\H{o}s' conjecture. Just this year,
Kwan, Sah, Sawhney, and Simkin proved Erd\H{o}s' conjecture. As for Steiner
systems with more general parameters, Glock, K\"{u}hn, Lo, and Osthus
conjectured the existence of high girth -Steiner systems. We prove the
approximate version of their conjecture.
This result follows from our general main results which concern finding
perfect or almost perfect matchings in a hypergraph avoiding a given set of
submatchings (which we view as a hypergraph where ). Our first
main result is a common generalization of the classical theorems of Pippenger
(for finding an almost perfect matching) and Ajtai, Koml\'os, Pintz, Spencer,
and Szemer\'edi (for finding an independent set in girth five hypergraphs).
More generally, we prove this for coloring and even list coloring, and also
generalize this further to when is a hypergraph with small codegrees (for
which high girth designs is a specific instance). Indeed, the coloring version
of our result even yields an almost partition of into approximate high
girth -Steiner systems.
Our main results also imply the existence of a perfect matching in a
bipartite hypergraph where the parts have slightly unbalanced degrees. This has
a number of applications; for example, it proves the existence of
pairwise disjoint list colorings in the setting of Kahn's theorem; it also
proves asymptotic versions of various rainbow matching results in the sparse
setting (where the number of times a color appears could be much smaller than
the number of colors) and even the existence of many pairwise disjoint rainbow
matchings in such circumstances.Comment: 52 page
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
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