509 research outputs found
Euler tours in hypergraphs
We show that a quasirandom -uniform hypergraph has a tight Euler tour
subject to the necessary condition that divides all vertex degrees. The
case when is complete confirms a conjecture of Chung, Diaconis and Graham
from 1989 on the existence of universal cycles for the -subsets of an
-set.Comment: version accepted for publication in Combinatoric
The Ramsey Number for 3-Uniform Tight Hypergraph Cycles
Let C(3)n denote the 3-uniform tight cycle, that is, the hypergraph with vertices v1, .â.â., vn and edges v1v2v3, v2v3v4, .â.â., vnâ1vnv1, vnv1v2. We prove that the smallest integer N = N(n) for which every redâblue colouring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of C(3)n is asymptotically equal to 4n/3 if n is divisible by 3, and 2n otherwise. The proof uses the regularity lemma for hypergraphs of Frankl and RĂśdl
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
Ramsey numbers of ordered graphs
An ordered graph is a pair where is a graph and
is a total ordering of its vertices. The ordered Ramsey number
is the minimum number such that every ordered
complete graph with vertices and with edges colored by two colors contains
a monochromatic copy of .
In contrast with the case of unordered graphs, we show that there are
arbitrarily large ordered matchings on vertices for which
is superpolynomial in . This implies that
ordered Ramsey numbers of the same graph can grow superpolynomially in the size
of the graph in one ordering and remain linear in another ordering.
We also prove that the ordered Ramsey number is
polynomial in the number of vertices of if the bandwidth of
is constant or if is an ordered graph of constant
degeneracy and constant interval chromatic number. The first result gives a
positive answer to a question of Conlon, Fox, Lee, and Sudakov.
For a few special classes of ordered paths, stars or matchings, we give
asymptotically tight bounds on their ordered Ramsey numbers. For so-called
monotone cycles we compute their ordered Ramsey numbers exactly. This result
implies exact formulas for geometric Ramsey numbers of cycles introduced by
K\'arolyi, Pach, T\'oth, and Valtr.Comment: 29 pages, 13 figures, to appear in Electronic Journal of
Combinatoric
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