5,049 research outputs found
Quasirandomness in hypergraphs
An -vertex graph of edge density is considered to be quasirandom
if it shares several important properties with the random graph . A
well-known theorem of Chung, Graham and Wilson states that many such `typical'
properties are asymptotically equivalent and, thus, a graph possessing one
such property automatically satisfies the others.
In recent years, work in this area has focused on uncovering more quasirandom
graph properties and on extending the known results to other discrete
structures. In the context of hypergraphs, however, one may consider several
different notions of quasirandomness. A complete description of these notions
has been provided recently by Towsner, who proved several central equivalences
using an analytic framework. We give short and purely combinatorial proofs of
the main equivalences in Towsner's result.Comment: 19 page
Embeddings and Ramsey numbers of sparse k-uniform hypergraphs
Chvatal, Roedl, Szemeredi and Trotter proved that the Ramsey numbers of
graphs of bounded maximum degree are linear in their order. In previous work,
we proved the same result for 3-uniform hypergraphs. Here we extend this result
to k-uniform hypergraphs, for any integer k > 3. As in the 3-uniform case, the
main new tool which we prove and use is an embedding lemma for k-uniform
hypergraphs of bounded maximum degree into suitable k-uniform `quasi-random'
hypergraphs.Comment: 24 pages, 2 figures. To appear in Combinatoric
Automatic enumeration of regular objects
We describe a framework for systematic enumeration of families combinatorial
structures which possess a certain regularity. More precisely, we describe how
to obtain the differential equations satisfied by their generating series.
These differential equations are then used to determine the initial counting
sequence and for asymptotic analysis. The key tool is the scalar product for
symmetric functions and that this operation preserves D-finiteness.Comment: Corrected for readability; To appear in the Journal of Integer
Sequence
Generalized multiplicities of edge ideals
We explore connections between the generalized multiplicities of square-free
monomial ideals and the combinatorial structure of the underlying hypergraphs
using methods of commutative algebra and polyhedral geometry. For instance, we
show the -multiplicity is multiplicative over the connected components of a
hypergraph, and we explicitly relate the -multiplicity of the edge ideal of
a properly connected uniform hypergraph to the Hilbert-Samuel multiplicity of
its special fiber ring. In addition, we provide general bounds for the
generalized multiplicities of the edge ideals and compute these invariants for
classes of uniform hypergraphs.Comment: 24 pages, 6 figures. The results of Theorem 4.6 and Theorem 9.2 are
now more general. To appear in Journal of Algebraic Combinatoric
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