38 research outputs found
Covering graphs by monochromatic trees and Helly-type results for hypergraphs
How many monochromatic paths, cycles or general trees does one need to cover
all vertices of a given -edge-coloured graph ? These problems were
introduced in the 1960s and were intensively studied by various researchers
over the last 50 years. In this paper, we establish a connection between this
problem and the following natural Helly-type question in hypergraphs. Roughly
speaking, this question asks for the maximum number of vertices needed to cover
all the edges of a hypergraph if it is known that any collection of a few
edges of has a small cover. We obtain quite accurate bounds for the
hypergraph problem and use them to give some unexpected answers to several
questions about covering graphs by monochromatic trees raised and studied by
Bal and DeBiasio, Kohayakawa, Mota and Schacht, Lang and Lo, and Gir\~ao,
Letzter and Sahasrabudhe.Comment: 20 pages including references plus 2 pages of an Appendi
Towards the Erd\H{o}s-Hajnal conjecture for -free graphs
The Erd\H{o}s-Hajnal conjecture is one of the most classical and well-known
problems in extremal and structural combinatorics dating back to 1977. It
asserts that in stark contrast to the case of a general -vertex graph if one
imposes even a little bit of structure on the graph, namely by forbidding a
fixed graph as an induced subgraph, instead of only being able to find a
polylogarithmic size clique or an independent set one can find one of
polynomial size. Despite being the focus of considerable attention over the
years the conjecture remains open. In this paper we improve the best known
lower bound of on this question, due to Erd\H{o}s
and Hajnal from 1989, in the smallest open case, namely when one forbids a
, the path on vertices. Namely, we show that any -free vertex
graph contains a clique or an independent set of size at least . Our methods also lead to the same improvement for an infinite
family of graphs
Towards the Erd\H{o}s-Gallai Cycle Decomposition Conjecture
In the 1960's, Erd\H{o}s and Gallai conjectured that the edges of any
-vertex graph can be decomposed into cycles and edges. We improve
upon the previous best bound of cycles and edges due to
Conlon, Fox and Sudakov, by showing an -vertex graph can always be
decomposed into cycles and edges, where is the
iterated logarithm function.Comment: Final version, accepted for publicatio
Minimum saturated families of sets
We call a family of subsets of -saturated if it
contains no pairwise disjoint sets, and moreover no set can be added to
while preserving this property (here ).
More than 40 years ago, Erd\H{o}s and Kleitman conjectured that an
-saturated family of subsets of has size at least . It is easy to show that every -saturated family has size at
least , but, as was mentioned by Frankl and Tokushige,
even obtaining a slightly better bound of , for some
fixed , seems difficult. In this note, we prove such a result,
showing that every -saturated family of subsets of has size at least
.
This lower bound is a consequence of a multipartite version of the problem,
in which we seek a lower bound on
where are families of subsets of ,
such that there are no pairwise disjoint sets, one from each family
, and furthermore no set can be added to any of the families
while preserving this property. We show that , which is tight e.g.\ by taking
to be empty, and letting the remaining families be the families
of all subsets of .Comment: 8 page
2-factors with k cycles in Hamiltonian graphs
A well known generalisation of Dirac's theorem states that if a graph on
vertices has minimum degree at least then contains a
-factor consisting of exactly cycles. This is easily seen to be tight in
terms of the bound on the minimum degree. However, if one assumes in addition
that is Hamiltonian it has been conjectured that the bound on the minimum
degree may be relaxed. This was indeed shown to be true by S\'ark\"ozy. In
subsequent papers, the minimum degree bound has been improved, most recently to
by DeBiasio, Ferrara, and Morris. On the other hand no
lower bounds close to this are known, and all papers on this topic ask whether
the minimum degree needs to be linear. We answer this question, by showing that
the required minimum degree for large Hamiltonian graphs to have a -factor
consisting of a fixed number of cycles is sublinear in Comment: 13 pages, 6 picture
Large independent sets from local considerations
The following natural problem was raised independently by Erd\H{o}s-Hajnal
and Linial-Rabinovich in the late 80's. How large must the independence number
of a graph be whose every vertices contain an independent
set of size ? In this paper we discuss new methods to attack this problem.
The first new approach, based on bounding Ramsey numbers of certain graphs,
allows us to improve previously best lower bounds due to Linial-Rabinovich,
Erd\H{o}s-Hajnal and Alon-Sudakov. As an example, we prove that any -vertex
graph having an independent set of size among every vertices has
. This confirms a conjecture of Erd\H{o}s and
Hajnal that should be at least and brings the
exponent half-way to the best possible value of . Our second approach
deals with upper bounds. It relies on a reduction of the original question to
the following natural extremal problem. What is the minimum possible value of
the -density of a graph on vertices having no independent set of size
? This allows us to improve previous upper bounds due to Linial-Rabinovich,
Krivelevich and Kostochka-Jancey. As part of our arguments we link the problem
of Erd\H{o}s-Hajnal and Linial-Rabinovich and our new extremal -density
problem to a number of other well-studied questions. This leads to many
interesting directions for future research.Comment: 26 pages in the main body, 7 figures, 3 appendice