1,304 research outputs found
On the chromatic number of a random hypergraph
We consider the problem of -colouring a random -uniform hypergraph with
vertices and edges, where , , remain constant as tends
to infinity. Achlioptas and Naor showed that the chromatic number of a random
graph in this setting, the case , must have one of two easily computable
values as tends to infinity. We give a complete generalisation of this
result to random uniform hypergraphs.Comment: 45 pages, 2 figures, revised versio
On the spectrum of hypergraphs
Here we study the spectral properties of an underlying weighted graph of a
non-uniform hypergraph by introducing different connectivity matrices, such as
adjacency, Laplacian and normalized Laplacian matrices. We show that different
structural properties of a hypergrpah, can be well studied using spectral
properties of these matrices. Connectivity of a hypergraph is also investigated
by the eigenvalues of these operators. Spectral radii of the same are bounded
by the degrees of a hypergraph. The diameter of a hypergraph is also bounded by
the eigenvalues of its connectivity matrices. We characterize different
properties of a regular hypergraph characterized by the spectrum. Strong
(vertex) chromatic number of a hypergraph is bounded by the eigenvalues.
Cheeger constant on a hypergraph is defined and we show that it can be bounded
by the smallest nontrivial eigenvalues of Laplacian matrix and normalized
Laplacian matrix, respectively, of a connected hypergraph. We also show an
approach to study random walk on a (non-uniform) hypergraph that can be
performed by analyzing the spectrum of transition probability operator which is
defined on that hypergraph. Ricci curvature on hypergraphs is introduced in two
different ways. We show that if the Laplace operator, , on a hypergraph
satisfies a curvature-dimension type inequality
with and then any non-zero eigenvalue of can be bounded below by . Eigenvalues of a normalized Laplacian operator defined on a connected
hypergraph can be bounded by the Ollivier's Ricci curvature of the hypergraph
H\"older-type inequalities and their applications to concentration and correlation bounds
Let be -valued random variables having a dependency
graph . We show that where is the -fold chromatic number
of . This inequality may be seen as a dependency-graph analogue of a
generalised H\"older inequality, due to Helmut Finner. Additionally, we provide
applications of H\"older-type inequalities to concentration and correlation
bounds for sums of weakly dependent random variables.Comment: 15 page
On some building blocks of hypergraphs: units, twin-units, regular, co-regular, and symmetric sets
Here, we introduce and investigate different building blocks, named units,
twin units, regular sets, symmetric sets, and co-regular sets in a hypergraph.
Our work shows that the presence of these building blocks leaves certain traces
in the spectrum and the corresponding eigenspaces of the connectivity operators
associated with the hypergraph. We also show that, conversely, some specific
footprints in the spectrum and in the corresponding eigenvectors retrace the
presence of some of these building blocks in the hypergraph. The hypergraph
remains invariant under the permutations among the vertices in some building
blocks. These vertices behave similarly, in random walks on the hypergraph and
play an important role in hypergraph automorphisms. Identifying similar
vertices in certain building blocks results in a smaller hypergraph that
contains some spectral information of the original hypergraph. The number of
specific building blocks provides an upper bound of the chromatic number of the
hypergraph. A pseudo metric is introduced to measure distances between vertices
in the hypergraph by using one of the building blocks. Here, we use the concept
of general connectivity operators of a hypergraph for our spectral study.Comment: Title is changed and some new section adde
On the Chromatic Thresholds of Hypergraphs
Let F be a family of r-uniform hypergraphs. The chromatic threshold of F is
the infimum of all non-negative reals c such that the subfamily of F comprising
hypergraphs H with minimum degree at least has bounded
chromatic number. This parameter has a long history for graphs (r=2), and in
this paper we begin its systematic study for hypergraphs.
{\L}uczak and Thomass\'e recently proved that the chromatic threshold of the
so-called near bipartite graphs is zero, and our main contribution is to
generalize this result to r-uniform hypergraphs. For this class of hypergraphs,
we also show that the exact Tur\'an number is achieved uniquely by the complete
(r+1)-partite hypergraph with nearly equal part sizes. This is one of very few
infinite families of nondegenerate hypergraphs whose Tur\'an number is
determined exactly. In an attempt to generalize Thomassen's result that the
chromatic threshold of triangle-free graphs is 1/3, we prove bounds for the
chromatic threshold of the family of 3-uniform hypergraphs not containing {abc,
abd, cde}, the so-called generalized triangle.
In order to prove upper bounds we introduce the concept of fiber bundles,
which can be thought of as a hypergraph analogue of directed graphs. This leads
to the notion of fiber bundle dimension, a structural property of fiber bundles
that is based on the idea of Vapnik-Chervonenkis dimension in hypergraphs. Our
lower bounds follow from explicit constructions, many of which use a hypergraph
analogue of the Kneser graph. Using methods from extremal set theory, we prove
that these Kneser hypergraphs have unbounded chromatic number. This generalizes
a result of Szemer\'edi for graphs and might be of independent interest. Many
open problems remain.Comment: 37 pages, 4 figure
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