186 research outputs found
Regularity of squarefree monomial ideals
We survey a number of recent studies of the Castelnuovo-Mumford regularity of
squarefree monomial ideals. Our focus is on bounds and exact values for the
regularity in terms of combinatorial data from associated simplicial complexes
and/or hypergraphs.Comment: 23 pages; survey paper; minor changes in V.
Overlap properties of geometric expanders
The {\em overlap number} of a finite -uniform hypergraph is
defined as the largest constant such that no matter how we map
the vertices of into , there is a point covered by at least a
-fraction of the simplices induced by the images of its hyperedges.
In~\cite{Gro2}, motivated by the search for an analogue of the notion of graph
expansion for higher dimensional simplicial complexes, it was asked whether or
not there exists a sequence of arbitrarily large
-uniform hypergraphs with bounded degree, for which . Using both random methods and explicit constructions, we answer this
question positively by constructing infinite families of -uniform
hypergraphs with bounded degree such that their overlap numbers are bounded
from below by a positive constant . We also show that, for every ,
the best value of the constant that can be achieved by such a
construction is asymptotically equal to the limit of the overlap numbers of the
complete -uniform hypergraphs with vertices, as
. For the proof of the latter statement, we establish the
following geometric partitioning result of independent interest. For any
and any , there exists satisfying the
following condition. For any , for any point and
for any finite Borel measure on with respect to which
every hyperplane has measure , there is a partition into measurable parts of equal measure such that all but
at most an -fraction of the -tuples
have the property that either all simplices with
one vertex in each contain or none of these simplices contain
Complexes of not -connected graphs
Complexes of (not) connected graphs, hypergraphs and their homology appear in
the construction of knot invariants given by V. Vassiliev. In this paper we
study the complexes of not -connected -hypergraphs on vertices. We
show that the complex of not -connected graphs has the homotopy type of a
wedge of spheres of dimension . This answers one of the
questions raised by Vassiliev in connection with knot invariants. For this case
the -action on the homology of the complex is also determined. For
complexes of not -connected -hypergraphs we provide a formula for the
generating function of the Euler characteristic, and we introduce certain
lattices of graphs that encode their topology. We also present partial results
for some other cases. In particular, we show that the complex of not
-connected graphs is Alexander dual to the complex of partial matchings
of the complete graph. For not -connected graphs we provide a formula
for the generating function of the Euler characteristic
Ramanujan Complexes and bounded degree topological expanders
Expander graphs have been a focus of attention in computer science in the
last four decades. In recent years a high dimensional theory of expanders is
emerging. There are several possible generalizations of the theory of expansion
to simplicial complexes, among them stand out coboundary expansion and
topological expanders. It is known that for every d there are unbounded degree
simplicial complexes of dimension d with these properties. However, a major
open problem, formulated by Gromov, is whether bounded degree high dimensional
expanders, according to these definitions, exist for d >= 2. We present an
explicit construction of bounded degree complexes of dimension d = 2 which are
high dimensional expanders. More precisely, our main result says that the
2-skeletons of the 3-dimensional Ramanujan complexes are topological expanders.
Assuming a conjecture of Serre on the congruence subgroup property, infinitely
many of them are also coboundary expanders.Comment: To appear in FOCS 201
Topological lower bounds for the chromatic number: A hierarchy
This paper is a study of ``topological'' lower bounds for the chromatic
number of a graph. Such a lower bound was first introduced by Lov\'asz in 1978,
in his famous proof of the \emph{Kneser conjecture} via Algebraic Topology.
This conjecture stated that the \emph{Kneser graph} \KG_{m,n}, the graph with
all -element subsets of as vertices and all pairs of
disjoint sets as edges, has chromatic number . Several other proofs
have since been published (by B\'ar\'any, Schrijver, Dolnikov, Sarkaria, Kriz,
Greene, and others), all of them based on some version of the Borsuk--Ulam
theorem, but otherwise quite different. Each can be extended to yield some
lower bound on the chromatic number of an arbitrary graph. (Indeed, we observe
that \emph{every} finite graph may be represented as a generalized Kneser
graph, to which the above bounds apply.)
We show that these bounds are almost linearly ordered by strength, the
strongest one being essentially Lov\'asz' original bound in terms of a
neighborhood complex. We also present and compare various definitions of a
\emph{box complex} of a graph (developing ideas of Alon, Frankl, and Lov\'asz
and of \kriz). A suitable box complex is equivalent to Lov\'asz' complex, but
the construction is simpler and functorial, mapping graphs with homomorphisms
to -spaces with -maps.Comment: 16 pages, 1 figure. Jahresbericht der DMV, to appea
Hypergraph expanders from Cayley graphs
We present a simple mechanism, which can be randomised, for constructing
sparse -uniform hypergraphs with strong expansion properties. These
hypergraphs are constructed using Cayley graphs over and have
vertex degree which is polylogarithmic in the number of vertices. Their
expansion properties, which are derived from the underlying Cayley graphs,
include analogues of vertex and edge expansion in graphs, rapid mixing of the
random walk on the edges of the skeleton graph, uniform distribution of edges
on large vertex subsets and the geometric overlap property.Comment: 13 page
Independence complexes and incidence graphs
We show that the independence complex of the incidence graph of a hypergraph is homotopy equivalent to the combinatorial Alexander dual of the independence complex of the hypergraph, generalizing a result of Csorba. As an application, we refine and generalize a result of Kawamura on a relation between the homotopy types of the independence complex and the edge covering complex of a graph
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