254 research outputs found
Covering complete partite hypergraphs by monochromatic components
A well-known special case of a conjecture attributed to Ryser states that
k-partite intersecting hypergraphs have transversals of at most k-1 vertices.
An equivalent form was formulated by Gy\'arf\'as: if the edges of a complete
graph K are colored with k colors then the vertex set of K can be covered by at
most k-1 sets, each connected in some color. It turned out that the analogue of
the conjecture for hypergraphs can be answered: Z. Kir\'aly proved that in
every k-coloring of the edges of the r-uniform complete hypergraph K^r (r >=
3), the vertex set of K^r can be covered by at most sets,
each connected in some color.
Here we investigate the analogue problem for complete r-uniform r-partite
hypergraphs. An edge coloring of a hypergraph is called spanning if every
vertex is incident to edges of any color used in the coloring. We propose the
following analogue of Ryser conjecture.
In every spanning (r+t)-coloring of the edges of a complete r-uniform
r-partite hypergraph, the vertex set can be covered by at most t+1 sets, each
connected in some color.
Our main result is that the conjecture is true for 1 <= t <= r-1. We also
prove a slightly weaker result for t >= r, namely that t+2 sets, each connected
in some color, are enough to cover the vertex set.
To build a bridge between complete r-uniform and complete r-uniform r-partite
hypergraphs, we introduce a new notion. A hypergraph is complete r-uniform
(r,l)-partite if it has all r-sets that intersect each partite class in at most
l vertices.
Extending our results achieved for l=1, we prove that for any r >= 3, 2 <= l
= 1+r-l, in every spanning k-coloring of the edges of a complete
r-uniform (r,l)-partite hypergraph, the vertex set can be covered by at most
1+\lfloor \frac{k-r+\ell-1}{\ell}\rfloor sets, each connected in some color.Comment: 14 page
On the tractability of some natural packing, covering and partitioning problems
In this paper we fix 7 types of undirected graphs: paths, paths with
prescribed endvertices, circuits, forests, spanning trees, (not necessarily
spanning) trees and cuts. Given an undirected graph and two "object
types" and chosen from the alternatives above, we
consider the following questions. \textbf{Packing problem:} can we find an
object of type and one of type in the edge set of
, so that they are edge-disjoint? \textbf{Partitioning problem:} can we
partition into an object of type and one of type ?
\textbf{Covering problem:} can we cover with an object of type
, and an object of type ? This framework includes 44
natural graph theoretic questions. Some of these problems were well-known
before, for example covering the edge-set of a graph with two spanning trees,
or finding an - path and an - path that are
edge-disjoint. However, many others were not, for example can we find an
- path and a spanning tree that are
edge-disjoint? Most of these previously unknown problems turned out to be
NP-complete, many of them even in planar graphs. This paper determines the
status of these 44 problems. For the NP-complete problems we also investigate
the planar version, for the polynomial problems we consider the matroidal
generalization (wherever this makes sense)
Problems and results on 1-cross intersecting set pair systems
The notion of cross intersecting set pair system of size ,
with and
, was introduced by Bollob\'as and it became an
important tool of extremal combinatorics. His classical result states that
if and for each .
Our central problem is to see how this bound changes with the additional
condition for . Such a system is called -cross
intersecting. We show that the maximum size of a -cross intersecting set
pair system is
-- at least for even, ,
-- equal to
if and ,
-- at most ,
-- asymptotically if is a linear hypergraph ( for ),
-- asymptotically if and are both linear
hypergraphs
On the Combinatorics of Projective Mappings
We consider composition sets of one-dimensional projective mappings and prove that small composition sets are closely related to Abelian subgroups
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