1,139 research outputs found
A note on the least number of edges of 3-uniform hypergraphs with upper chromatic number 2
AbstractThe upper chromatic number χ¯(H) of a hypergraph H=(X,E) is the maximum number k for which there exists a partition of X into non-empty subsets X=X1∪X2∪⋯∪Xk such that for each edge at least two vertices lie in one of the partite sets. We prove that for every n⩾3 there exists a 3-uniform hypergraph with n vertices, upper chromatic number 2 and ⌈n(n-2)/3⌉ edges which implies that a corresponding bound proved in [K. Diao, P. Zhao, H. Zhou, About the upper chromatic number of a co-hypergraph, Discrete Math. 220 (2000) 67–73] is best-possible
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
Coloring d-Embeddable k-Uniform Hypergraphs
This paper extends the scenario of the Four Color Theorem in the following
way. Let H(d,k) be the set of all k-uniform hypergraphs that can be (linearly)
embedded into R^d. We investigate lower and upper bounds on the maximum (weak
and strong) chromatic number of hypergraphs in H(d,k). For example, we can
prove that for d>2 there are hypergraphs in H(2d-3,d) on n vertices whose weak
chromatic number is Omega(log n/log log n), whereas the weak chromatic number
for n-vertex hypergraphs in H(d,d) is bounded by O(n^((d-2)/(d-1))) for d>2.Comment: 18 page
Chromatic Ramsey number of acyclic hypergraphs
Suppose that is an acyclic -uniform hypergraph, with . We
define the (-color) chromatic Ramsey number as the smallest
with the following property: if the edges of any -chromatic -uniform
hypergraph are colored with colors in any manner, there is a monochromatic
copy of . We observe that is well defined and where
is the -color Ramsey number of . We give linear upper bounds
for when T is a matching or star, proving that for , and where
and are, respectively, the -uniform matching and star with
edges.
The general bounds are improved for -uniform hypergraphs. We prove that
, extending a special case of Alon-Frankl-Lov\'asz' theorem.
We also prove that , which is sharp for . This is
a corollary of a more general result. We define as the 1-intersection
graph of , whose vertices represent hyperedges and whose edges represent
intersections of hyperedges in exactly one vertex. We prove that for any -uniform hypergraph (assuming ). The proof uses the list coloring version of Brooks' theorem.Comment: 10 page
On generalized Kneser hypergraph colorings
In Ziegler (2002), the second author presented a lower bound for the
chromatic numbers of hypergraphs \KG{r}{\pmb s}{\calS}, "generalized
-uniform Kneser hypergraphs with intersection multiplicities ." It
generalized previous lower bounds by Kriz (1992/2000) for the case without intersection multiplicities, and by Sarkaria (1990) for
\calS=\tbinom{[n]}k. Here we discuss subtleties and difficulties that arise
for intersection multiplicities :
1. In the presence of intersection multiplicities, there are two different
versions of a "Kneser hypergraph," depending on whether one admits hypergraph
edges that are multisets rather than sets. We show that the chromatic numbers
are substantially different for the two concepts of hypergraphs. The lower
bounds of Sarkaria (1990) and Ziegler (2002) apply only to the multiset
version.
2. The reductions to the case of prime in the proofs Sarkaria and by
Ziegler work only if the intersection multiplicities are strictly smaller than
the largest prime factor of . Currently we have no valid proof for the lower
bound result in the other cases.
We also show that all uniform hypergraphs without multiset edges can be
represented as generalized Kneser hypergraphs.Comment: 9 pages; added examples in Section 2; added reference ([11]),
corrected minor typos; to appear in J. Combinatorial Theory, Series
Generalisation : graphs and colourings
The interaction between practice and theory in mathematics is a central theme. Many mathematical structures and theories result from the formalisation of a real problem. Graph Theory is rich with such examples. The graph structure itself was formalised by Leonard Euler in the quest to solve the problem of the Bridges of Königsberg. Once a structure is formalised, and results are proven, the mathematician seeks to generalise. This can be considered as one of the main praxis in mathematics. The idea of generalisation will be illustrated through graph colouring. This idea also results from a classic problem, in which it was well known by topographers that four colours suffice to colour any map such that no countries sharing a border receive the same colour. The proof of this theorem eluded mathematicians for centuries and was proven in 1976. Generalisation of graphs to hypergraphs, and variations on the colouring theme will be discussed, as well as applications in other disciplines.peer-reviewe
- …