2,224 research outputs found
K 4-free subgraphs of random graphs revisited
In Combinatorica 17(2), 1997, Kohayakawa, Ćuczak and Rödl state a conjecture which has several implications for random graphs. If the conjecture is true, then, for example, an application of a version of SzemerĂ©di's regularity lemma for sparse graphs yields an estimation of the maximal number of edges in an H-free subgraph of a random graph G n, p . In fact, the conjecture may be seen as a probabilistic embedding lemma for partitions guaranteed by a version of SzemerĂ©di's regularity lemma for sparse graphs. In this paper we verify the conjecture for H = K 4, thereby providing a conceptually simple proof for the main result in the paper cited abov
Decomposition of bounded degree graphs into -free subgraphs
We prove that every graph with maximum degree admits a partition of
its edges into parts (as ) none of which
contains as a subgraph. This bound is sharp up to a constant factor. Our
proof uses an iterated random colouring procedure.Comment: 8 pages; to appear in European Journal of Combinatoric
Finite graphs and amenability
Hyperfiniteness or amenability of measurable equivalence relations and group
actions has been studied for almost fifty years. Recently, unexpected
applications of hyperfiniteness were found in computer science in the context
of testability of graph properties. In this paper we propose a unified approach
to hyperfiniteness. We establish some new results and give new proofs of
theorems of Schramm, Lov\'asz, Newman-Sohler and Ornstein-Weiss
Combinatorial theorems relative to a random set
We describe recent advances in the study of random analogues of combinatorial
theorems.Comment: 26 pages. Submitted to Proceedings of the ICM 201
All reducts of the random graph are model-complete
We study locally closed transformation monoids which contain the automorphism
group of the random graph. We show that such a transformation monoid is locally
generated by the permutations in the monoid, or contains a constant operation,
or contains an operation that maps the random graph injectively to an induced
subgraph which is a clique or an independent set. As a corollary, our
techniques yield a new proof of Simon Thomas' classification of the five closed
supergroups of the automorphism group of the random graph; our proof uses
different Ramsey-theoretic tools than the one given by Thomas, and is perhaps
more straightforward. Since the monoids under consideration are endomorphism
monoids of relational structures definable in the random graph, we are able to
draw several model-theoretic corollaries: One consequence of our result is that
all structures with a first-order definition in the random graph are
model-complete. Moreover, we obtain a classification of these structures up to
existential interdefinability.Comment: Technical report not intended for publication in a journal. Subsumed
by the more recent article 1003.4030. Length 14 pages
Beyond Triangles: A Distributed Framework for Estimating 3-profiles of Large Graphs
We study the problem of approximating the -profile of a large graph.
-profiles are generalizations of triangle counts that specify the number of
times a small graph appears as an induced subgraph of a large graph. Our
algorithm uses the novel concept of -profile sparsifiers: sparse graphs that
can be used to approximate the full -profile counts for a given large graph.
Further, we study the problem of estimating local and ego -profiles, two
graph quantities that characterize the local neighborhood of each vertex of a
graph.
Our algorithm is distributed and operates as a vertex program over the
GraphLab PowerGraph framework. We introduce the concept of edge pivoting which
allows us to collect -hop information without maintaining an explicit
-hop neighborhood list at each vertex. This enables the computation of all
the local -profiles in parallel with minimal communication.
We test out implementation in several experiments scaling up to cores
on Amazon EC2. We find that our algorithm can estimate the -profile of a
graph in approximately the same time as triangle counting. For the harder
problem of ego -profiles, we introduce an algorithm that can estimate
profiles of hundreds of thousands of vertices in parallel, in the timescale of
minutes.Comment: To appear in part at KDD'1
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