477 research outputs found
Generalized quasirandom graphs
AbstractWe prove that if a sequence of graphs has (asymptotically) the same distribution of small subgraphs as a generalized random graph modeled on a fixed weighted graph H, then these graphs have a structure that is asymptotically the same as the structure of H. Furthermore, it suffices to require this for a finite number of subgraphs, whose number and size is bounded by a function of |V(H)|
Forcing Generalized Quasirandom Graphs Efficiently
We study generalized quasirandom graphs whose vertex set consists of
parts (of not necessarily the same sizes) with edges within each part and
between each pair of parts distributed quasirandomly; such graphs correspond to
the stochastic block model studied in statistics and network science. Lov\'asz
and S\'os showed that the structure of such graphs is forced by homomorphism
densities of graphs with at most vertices; subsequently, Lov\'asz
refined the argument to show that graphs with vertices suffice. Our
results imply that the structure of generalized quasirandom graphs with parts is forced by homomorphism densities of graphs with at most
vertices, and, if vertices in distinct parts have distinct degrees, then
vertices suffice. The latter improves the bound of due to Spencer
Perfect Packings in Quasirandom Hypergraphs II
For each of the notions of hypergraph quasirandomness that have been studied,
we identify a large class of hypergraphs F so that every quasirandom hypergraph
H admits a perfect F-packing. An informal statement of a special case of our
general result for 3-uniform hypergraphs is as follows. Fix an integer r >= 4
and 0<p<1. Suppose that H is an n-vertex triple system with r|n and the
following two properties:
* for every graph G with V(G)=V(H), at least p proportion of the triangles in
G are also edges of H,
* for every vertex x of H, the link graph of x is a quasirandom graph with
density at least p.
Then H has a perfect -packing. Moreover, we show that neither
hypotheses above can be weakened, so in this sense our result is tight. A
similar conclusion for this special case can be proved by Keevash's hypergraph
blowup lemma, with a slightly stronger hypothesis on H.Comment: 17 page
Quasirandom permutations are characterized by 4-point densities
For permutations π and τ of lengths |π|≤|τ| , let t(π,τ) be the probability that the restriction of τ to a random |π| -point set is (order) isomorphic to π . We show that every sequence {τj} of permutations such that |τj|→∞ and t(π,τj)→1/4! for every 4-point permutation π is quasirandom (that is, t(π,τj)→1/|π|! for every π ). This answers a question posed by Graham
An approximate version of Sidorenko's conjecture
A beautiful conjecture of Erd\H{o}s-Simonovits and Sidorenko states that if H
is a bipartite graph, then the random graph with edge density p has in
expectation asymptotically the minimum number of copies of H over all graphs of
the same order and edge density. This conjecture also has an equivalent
analytic form and has connections to a broad range of topics, such as matrix
theory, Markov chains, graph limits, and quasirandomness. Here we prove the
conjecture if H has a vertex complete to the other part, and deduce an
approximate version of the conjecture for all H. Furthermore, for a large class
of bipartite graphs, we prove a stronger stability result which answers a
question of Chung, Graham, and Wilson on quasirandomness for these graphs.Comment: 12 page
Quasirandomness in hypergraphs
An -vertex graph of edge density is considered to be quasirandom
if it shares several important properties with the random graph . A
well-known theorem of Chung, Graham and Wilson states that many such `typical'
properties are asymptotically equivalent and, thus, a graph possessing one
such property automatically satisfies the others.
In recent years, work in this area has focused on uncovering more quasirandom
graph properties and on extending the known results to other discrete
structures. In the context of hypergraphs, however, one may consider several
different notions of quasirandomness. A complete description of these notions
has been provided recently by Towsner, who proved several central equivalences
using an analytic framework. We give short and purely combinatorial proofs of
the main equivalences in Towsner's result.Comment: 19 page
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