21 research outputs found
Rainbow Generalizations of Ramsey Theory - A Dynamic Survey
In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs
Rainbow Generalizations of Ramsey Theory - A Dynamic Survey
In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs
Rainbow Generalizations of Ramsey Theory - A Dynamic Survey
In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs
Sparse Cuts in Hypergraphs from Random Walks on Simplicial Complexes
There are a lot of recent works on generalizing the spectral theory of graphs
and graph partitioning to hypergraphs. There have been two broad directions
toward this goal. One generalizes the notion of graph conductance to hypergraph
conductance [LM16, CLTZ18]. In the second approach one can view a hypergraph as
a simplicial complex and study its various topological properties [LM06, MW09,
DKW16, PR17] and spectral properties [KM17, DK17, KO18a, KO18b, Opp20].
In this work, we attempt to bridge these two directions of study by relating
the spectrum of up-down walks and swap-walks on the simplicial complex to
hypergraph expansion. In surprising contrast to random-walks on graphs, we show
that the spectral gap of swap-walks can not be used to infer any bounds on
hypergraph conductance. For the up-down walks, we show that spectral gap of
walks between levels satisfying can not be used to bound
hypergraph expansion. We give a Cheeger-like inequality relating the spectral
of walks between level 1 and to hypergraph expansion.
Finally, we also give a construction to show that the well-studied notion of
link expansion in simplicial complexes can not be used to bound hypergraph
expansion in a Cheeger like manner.Comment: 25 page
Extremal and Ramsey Type Questions for Graphs and Ordered Graphs
In this thesis we study graphs and ordered graphs from an extremal point of view. In the first part of the thesis we prove that there are ordered forests H and ordered graphs of arbitrarily large chromatic number not containing such H as an ordered subgraph. In the second part we study pairs of graphs that have the same set of Ramsey graphs. We support a negative answer to the question whether there are pairs of non-isomorphic connected graphs that have this property. Finally we initiate the study of minimal ordered Ramsey graphs. For large families of ordered graphs we determine whether their members have finitely or infinitely many minimal ordered Ramsey graphs
Unsplittable coverings in the plane
A system of sets forms an {\em -fold covering} of a set if every point
of belongs to at least of its members. A -fold covering is called a
{\em covering}. The problem of splitting multiple coverings into several
coverings was motivated by classical density estimates for {\em sphere
packings} as well as by the {\em planar sensor cover problem}. It has been the
prevailing conjecture for 35 years (settled in many special cases) that for
every plane convex body , there exists a constant such that every
-fold covering of the plane with translates of splits into
coverings. In the present paper, it is proved that this conjecture is false for
the unit disk. The proof can be generalized to construct, for every , an
unsplittable -fold covering of the plane with translates of any open convex
body which has a smooth boundary with everywhere {\em positive curvature}.
Somewhat surprisingly, {\em unbounded} open convex sets do not misbehave,
they satisfy the conjecture: every -fold covering of any region of the plane
by translates of such a set splits into two coverings. To establish this
result, we prove a general coloring theorem for hypergraphs of a special type:
{\em shift-chains}. We also show that there is a constant such that, for
any positive integer , every -fold covering of a region with unit disks
splits into two coverings, provided that every point is covered by {\em at
most} sets
In the shadows of a hypergraph: looking for associated primes of powers of square-free monomial ideals
The aim of this paper is to study the associated primes of powers of square-free monomial ideals. Each square-free monomial ideal corresponds uniquely to a finite simple hypergraph via the cover ideal construction, and vice versa. Let H be a finite simple hypergraph and J(H) the cover ideal of H. We define the shadows of hypergraph, H, described as a collection of smaller hypergraphs related to H under some conditions. We then investigate how the shadows of H preserve information about the associated primes of the powers of J(H). Finally, we apply our findings on shadows to study the persistence property of square-free monomial ideals and construct some examples exhibiting failure of containment