49,235 research outputs found
Partitions of graphs into small and large sets
Let be a graph on vertices. We call a subset of the vertex set
\emph{-small} if, for every vertex , . A subset is called \emph{-large} if, for every vertex
, . Moreover, we denote by the
minimum integer such that there is a partition of into -small
sets, and by the minimum integer such that there is a
partition of into -large sets. In this paper, we will show tight
connections between -small sets, respectively -large sets, and the
-independence number, the clique number and the chromatic number of a graph.
We shall develop greedy algorithms to compute in linear time both
and and prove various sharp inequalities
concerning these parameters, which we will use to obtain refinements of the
Caro-Wei Theorem, the Tur\'an Theorem and the Hansen-Zheng Theorem among other
things.Comment: 21 page
Szemer\'edi's Regularity Lemma for matrices and sparse graphs
Szemer\'edi's Regularity Lemma is an important tool for analyzing the
structure of dense graphs. There are versions of the Regularity Lemma for
sparse graphs, but these only apply when the graph satisfies some local density
condition. In this paper, we prove a sparse Regularity Lemma that holds for all
graphs. More generally, we give a Regularity Lemma that holds for arbitrary
real matrices
SMART-KG: Hybrid Shipping for SPARQL Querying on the Web
While Linked Data (LD) provides standards for publishing (RDF) and (SPARQL) querying Knowledge Graphs (KGs) on the Web, serving, accessing and processing such open, decentralized KGs is often practically impossible, as query timeouts on publicly available SPARQL endpoints show. Alternative solutions such as Triple Pattern Fragments (TPF) attempt to tackle the problem of availability by pushing query processing workload to the client side, but suffer from unnecessary transfer of irrelevant data on complex queries with large intermediate results. In this paper we present smart-KG, a novel approach to share the load between servers and clients, while significantly reducing data transfer volume, by combining TPF with shipping compressed KG partitions. Our evaluations show that outperforms state-of-the-art client-side solutions and increases server-side availability towards more cost-effective and balanced hosting of open and decentralized KGs.Series: Working Papers on Information Systems, Information Business and Operation
Embedding into bipartite graphs
The conjecture of Bollob\'as and Koml\'os, recently proved by B\"ottcher,
Schacht, and Taraz [Math. Ann. 343(1), 175--205, 2009], implies that for any
, every balanced bipartite graph on vertices with bounded degree
and sublinear bandwidth appears as a subgraph of any -vertex graph with
minimum degree , provided that is sufficiently large. We show
that this threshold can be cut in half to an essentially best-possible minimum
degree of when we have the additional structural
information of the host graph being balanced bipartite. This complements
results of Zhao [to appear in SIAM J. Discrete Math.], as well as Hladk\'y and
Schacht [to appear in SIAM J. Discrete Math.], who determined a corresponding
minimum degree threshold for -factors, with and fixed.
Moreover, it implies that the set of Hamilton cycles of is a generating
system for its cycle space.Comment: 16 pages, 2 figure
Convergent Sequences of Dense Graphs I: Subgraph Frequencies, Metric Properties and Testing
We consider sequences of graphs and define various notions of convergence
related to these sequences: ``left convergence'' defined in terms of the
densities of homomorphisms from small graphs into the graphs of the sequence,
and ``right convergence'' defined in terms of the densities of homomorphisms
from the graphs of the sequence into small graphs; and convergence in a
suitably defined metric.
In Part I of this series, we show that left convergence is equivalent to
convergence in metric, both for simple graphs, and for graphs with nodeweights
and edgeweights. One of the main steps here is the introduction of a
cut-distance comparing graphs, not necessarily of the same size. We also show
how these notions of convergence provide natural formulations of Szemeredi
partitions, sampling and testing of large graphs.Comment: 57 pages. See also http://research.microsoft.com/~borgs/. This
version differs from an earlier version from May 2006 in the organization of
the sections, but is otherwise almost identica
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