4,711 research outputs found
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
Efficient Construction of Probabilistic Tree Embeddings
In this paper we describe an algorithm that embeds a graph metric
on an undirected weighted graph into a distribution of tree metrics
such that for every pair , and
. Such embeddings have
proved highly useful in designing fast approximation algorithms, as many hard
problems on graphs are easy to solve on tree instances. For a graph with
vertices and edges, our algorithm runs in time with high
probability, which improves the previous upper bound of shown by
Mendel et al.\,in 2009.
The key component of our algorithm is a new approximate single-source
shortest-path algorithm, which implements the priority queue with a new data
structure, the "bucket-tree structure". The algorithm has three properties: it
only requires linear time in the number of edges in the input graph; the
computed distances have a distance preserving property; and when computing the
shortest-paths to the -nearest vertices from the source, it only requires to
visit these vertices and their edge lists. These properties are essential to
guarantee the correctness and the stated time bound.
Using this shortest-path algorithm, we show how to generate an intermediate
structure, the approximate dominance sequences of the input graph, in time, and further propose a simple yet efficient algorithm to converted
this sequence to a tree embedding in time, both with high
probability. Combining the three subroutines gives the stated time bound of the
algorithm.
Then we show that this efficient construction can facilitate some
applications. We proved that FRT trees (the generated tree embedding) are
Ramsey partitions with asymptotically tight bound, so the construction of a
series of distance oracles can be accelerated
Partitioning random graphs into monochromatic components
Erd\H{o}s, Gy\'arf\'as, and Pyber (1991) conjectured that every -colored
complete graph can be partitioned into at most monochromatic components;
this is a strengthening of a conjecture of Lov\'asz (1975) in which the
components are only required to form a cover. An important partial result of
Haxell and Kohayakawa (1995) shows that a partition into monochromatic
components is possible for sufficiently large -colored complete graphs.
We start by extending Haxell and Kohayakawa's result to graphs with large
minimum degree, then we provide some partial analogs of their result for random
graphs. In particular, we show that if , then a.a.s. in every -coloring of there exists
a partition into two monochromatic components, and for if , then a.a.s. there exists an -coloring
of such that there does not exist a cover with a bounded number of
components. Finally, we consider a random graph version of a classic result of
Gy\'arf\'as (1977) about large monochromatic components in -colored complete
graphs. We show that if , then a.a.s. in every
-coloring of there exists a monochromatic component of order at
least .Comment: 27 pages, 2 figures. Appears in Electronic Journal of Combinatorics
Volume 24, Issue 1 (2017) Paper #P1.1
Spanning embeddings of arrangeable graphs with sublinear bandwidth
The Bandwidth Theorem of B\"ottcher, Schacht and Taraz [Mathematische Annalen
343 (1), 175-205] gives minimum degree conditions for the containment of
spanning graphs H with small bandwidth and bounded maximum degree. We
generalise this result to a-arrangeable graphs H with \Delta(H)<sqrt(n)/log(n),
where n is the number of vertices of H.
Our result implies that sufficiently large n-vertex graphs G with minimum
degree at least (3/4+\gamma)n contain almost all planar graphs on n vertices as
subgraphs. Using techniques developed by Allen, Brightwell and Skokan
[Combinatorica, to appear] we can also apply our methods to show that almost
all planar graphs H have Ramsey number at most 12|H|. We obtain corresponding
results for graphs embeddable on different orientable surfaces.Comment: 20 page
Embedding Metrics into Ultrametrics and Graphs into Spanning Trees with Constant Average Distortion
This paper addresses the basic question of how well can a tree approximate
distances of a metric space or a graph. Given a graph, the problem of
constructing a spanning tree in a graph which strongly preserves distances in
the graph is a fundamental problem in network design. We present scaling
distortion embeddings where the distortion scales as a function of ,
with the guarantee that for each the distortion of a fraction
of all pairs is bounded accordingly. Such a bound implies, in
particular, that the \emph{average distortion} and -distortions are
small. Specifically, our embeddings have \emph{constant} average distortion and
-distortion. This follows from the following
results: we prove that any metric space embeds into an ultrametric with scaling
distortion . For the graph setting we prove that any
weighted graph contains a spanning tree with scaling distortion
. These bounds are tight even for embedding in arbitrary
trees.
For probabilistic embedding into spanning trees we prove a scaling distortion
of , which implies \emph{constant}
-distortion for every fixed .Comment: Extended abstrat apears in SODA 200
Properly coloured copies and rainbow copies of large graphs with small maximum degree
Let G be a graph on n vertices with maximum degree D. We use the Lov\'asz
local lemma to show the following two results about colourings c of the edges
of the complete graph K_n. If for each vertex v of K_n the colouring c assigns
each colour to at most (n-2)/22.4D^2 edges emanating from v, then there is a
copy of G in K_n which is properly edge-coloured by c. This improves on a
result of Alon, Jiang, Miller, and Pritikin [Random Struct. Algorithms 23(4),
409-433, 2003]. On the other hand, if c assigns each colour to at most n/51D^2
edges of K_n, then there is a copy of G in K_n such that each edge of G
receives a different colour from c. This proves a conjecture of Frieze and
Krivelevich [Electron. J. Comb. 15(1), R59, 2008]. Our proofs rely on a
framework developed by Lu and Sz\'ekely [Electron. J. Comb. 14(1), R63, 2007]
for applying the local lemma to random injections. In order to improve the
constants in our results we use a version of the local lemma due to Bissacot,
Fern\'andez, Procacci, and Scoppola [preprint, arXiv:0910.1824].Comment: 9 page
Positional Games
Positional games are a branch of combinatorics, researching a variety of
two-player games, ranging from popular recreational games such as Tic-Tac-Toe
and Hex, to purely abstract games played on graphs and hypergraphs. It is
closely connected to many other combinatorial disciplines such as Ramsey
theory, extremal graph and set theory, probabilistic combinatorics, and to
computer science. We survey the basic notions of the field, its approaches and
tools, as well as numerous recent advances, standing open problems and
promising research directions.Comment: Submitted to Proceedings of the ICM 201
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