10,302 research outputs found
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
Extremal results in sparse pseudorandom graphs
Szemer\'edi's regularity lemma is a fundamental tool in extremal
combinatorics. However, the original version is only helpful in studying dense
graphs. In the 1990s, Kohayakawa and R\"odl proved an analogue of Szemer\'edi's
regularity lemma for sparse graphs as part of a general program toward
extending extremal results to sparse graphs. Many of the key applications of
Szemer\'edi's regularity lemma use an associated counting lemma. In order to
prove extensions of these results which also apply to sparse graphs, it
remained a well-known open problem to prove a counting lemma in sparse graphs.
The main advance of this paper lies in a new counting lemma, proved following
the functional approach of Gowers, which complements the sparse regularity
lemma of Kohayakawa and R\"odl, allowing us to count small graphs in regular
subgraphs of a sufficiently pseudorandom graph. We use this to prove sparse
extensions of several well-known combinatorial theorems, including the removal
lemmas for graphs and groups, the Erd\H{o}s-Stone-Simonovits theorem and
Ramsey's theorem. These results extend and improve upon a substantial body of
previous work.Comment: 70 pages, accepted for publication in Adv. Mat
Which point sets admit a k-angulation?
For k >= 3, a k-angulation is a 2-connected plane graph in which every
internal face is a k-gon. We say that a point set P admits a plane graph G if
there is a straight-line drawing of G that maps V(G) onto P and has the same
facial cycles and outer face as G. We investigate the conditions under which a
point set P admits a k-angulation and find that, for sets containing at least
2k^2 points, the only obstructions are those that follow from Euler's formula.Comment: 13 pages, 7 figure
Some results on triangle partitions
We show that there exist efficient algorithms for the triangle packing
problem in colored permutation graphs, complete multipartite graphs,
distance-hereditary graphs, k-modular permutation graphs and complements of
k-partite graphs (when k is fixed). We show that there is an efficient
algorithm for C_4-packing on bipartite permutation graphs and we show that
C_4-packing on bipartite graphs is NP-complete. We characterize the cobipartite
graphs that have a triangle partition
Partitioning Graph Drawings and Triangulated Simple Polygons into Greedily Routable Regions
A greedily routable region (GRR) is a closed subset of , in
which each destination point can be reached from each starting point by
choosing the direction with maximum reduction of the distance to the
destination in each point of the path.
Recently, Tan and Kermarrec proposed a geographic routing protocol for dense
wireless sensor networks based on decomposing the network area into a small
number of interior-disjoint GRRs. They showed that minimum decomposition is
NP-hard for polygons with holes.
We consider minimum GRR decomposition for plane straight-line drawings of
graphs. Here, GRRs coincide with self-approaching drawings of trees, a drawing
style which has become a popular research topic in graph drawing. We show that
minimum decomposition is still NP-hard for graphs with cycles, but can be
solved optimally for trees in polynomial time. Additionally, we give a
2-approximation for simple polygons, if a given triangulation has to be
respected.Comment: full version of a paper appearing in ISAAC 201
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