449 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
On the minimum degree of minimal Ramsey graphs for multiple colours
A graph G is r-Ramsey for a graph H, denoted by G\rightarrow (H)_r, if every
r-colouring of the edges of G contains a monochromatic copy of H. The graph G
is called r-Ramsey-minimal for H if it is r-Ramsey for H but no proper subgraph
of G possesses this property. Let s_r(H) denote the smallest minimum degree of
G over all graphs G that are r-Ramsey-minimal for H. The study of the parameter
s_2 was initiated by Burr, Erd\H{o}s, and Lov\'{a}sz in 1976 when they showed
that for the clique s_2(K_k)=(k-1)^2. In this paper, we study the dependency of
s_r(K_k) on r and show that, under the condition that k is constant, s_r(K_k) =
r^2 polylog r. We also give an upper bound on s_r(K_k) which is polynomial in
both r and k, and we determine s_r(K_3) up to a factor of log r
Path Coupling Using Stopping Times and Counting Independent Sets and Colourings in Hypergraphs
We give a new method for analysing the mixing time of a Markov chain using
path coupling with stopping times. We apply this approach to two hypergraph
problems. We show that the Glauber dynamics for independent sets in a
hypergraph mixes rapidly as long as the maximum degree Delta of a vertex and
the minimum size m of an edge satisfy m>= 2Delta+1. We also show that the
Glauber dynamics for proper q-colourings of a hypergraph mixes rapidly if m>= 4
and q > Delta, and if m=3 and q>=1.65Delta. We give related results on the
hardness of exact and approximate counting for both problems.Comment: Simpler proof of main theorem. Improved bound on mixing time. 19
page
Embedding large subgraphs into dense graphs
What conditions ensure that a graph G contains some given spanning subgraph
H? The most famous examples of results of this kind are probably Dirac's
theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect
matchings are generalized by perfect F-packings, where instead of covering all
the vertices of G by disjoint edges, we want to cover G by disjoint copies of a
(small) graph F. It is unlikely that there is a characterization of all graphs
G which contain a perfect F-packing, so as in the case of Dirac's theorem it
makes sense to study conditions on the minimum degree of G which guarantee a
perfect F-packing.
The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy
and Szemeredi have proved to be powerful tools in attacking such problems and
quite recently, several long-standing problems and conjectures in the area have
been solved using these. In this survey, we give an outline of recent progress
(with our main emphasis on F-packings, Hamiltonicity problems and tree
embeddings) and describe some of the methods involved
The history of degenerate (bipartite) extremal graph problems
This paper is a survey on Extremal Graph Theory, primarily focusing on the
case when one of the excluded graphs is bipartite. On one hand we give an
introduction to this field and also describe many important results, methods,
problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version
of our survey presented in Erdos 100. In this version 2 only a citation was
complete
Jigsaw percolation on random hypergraphs
The jigsaw percolation process on graphs was introduced by Brummitt,
Chatterjee, Dey, and Sivakoff as a model of collaborative solutions of puzzles
in social networks. Percolation in this process may be viewed as the joint
connectedness of two graphs on a common vertex set. Our aim is to extend a
result of Bollob\'as, Riordan, Slivken, and Smith concerning this process to
hypergraphs for a variety of possible definitions of connectedness. In
particular, we determine the asymptotic order of the critical threshold
probability for percolation when both hypergraphs are chosen binomially at
random.Comment: 17 page
The Erd\H{o}s-Rothschild problem on edge-colourings with forbidden monochromatic cliques
Let be a sequence of natural numbers. For a
graph , let denote the number of colourings of the edges
of with colours such that, for every , the
edges of colour contain no clique of order . Write
to denote the maximum of over all graphs on vertices.
This problem was first considered by Erd\H{o}s and Rothschild in 1974, but it
has been solved only for a very small number of non-trivial cases.
We prove that, for every and , there is a complete
multipartite graph on vertices with . Also, for every we construct a finite
optimisation problem whose maximum is equal to the limit of as tends to infinity. Our final result is a
stability theorem for complete multipartite graphs , describing the
asymptotic structure of such with in terms of solutions to the optimisation problem.Comment: 16 pages, to appear in Math. Proc. Cambridge Phil. So
Rainbow Turán Problems
For a fixed graph H, we define the rainbow Turán number ex^*(n,H) to be the maximum number of edges in a graph on n vertices that has a proper edge-colouring with no rainbow H. Recall that the (ordinary) Turán number ex(n,H) is the maximum number of edges in a graph on n vertices that does not contain a copy of H. For any non-bipartite H we show that ex^*(n,H)=(1+o(1))ex(n,H), and if H is colour-critical we show that ex^{*}(n,H)=ex(n,H). When H is the complete bipartite graph K_{s,t} with s ≤ t we show ex^*(n,K_{s,t}) = O(n^{2-1/s}), which matches the known bounds for ex(n,K_{s,t}) up to a constant. We also study the rainbow Turán problem for even cycles, and in particular prove the bound ex^*(n,C_6) = O(n^{4/3}), which is of the correct order of magnitude
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