9,068 research outputs found
Dense Subgraphs in Random Graphs
For a constant and a graph , let be
the largest integer for which there exists a -vertex subgraph of
with at least edges. We show that if then
is concentrated on a set of two integers. More
precisely, with
,
we show that is one of the two integers closest to
, with high probability.
While this situation parallels that of cliques in random graphs, a new
technique is required to handle the more complicated ways in which these
"quasi-cliques" may overlap
Finding Connected Dense -Subgraphs
Given a connected graph on vertices and a positive integer ,
a subgraph of on vertices is called a -subgraph in . We design
combinatorial approximation algorithms for finding a connected -subgraph in
such that its density is at least a factor
of the density of the densest -subgraph
in (which is not necessarily connected). These particularly provide the
first non-trivial approximations for the densest connected -subgraph problem
on general graphs
Finding cliques and dense subgraphs using edge queries
We consider the problem of finding a large clique in an Erd\H{o}s--R\'enyi
random graph where we are allowed unbounded computational time but can only
query a limited number of edges. Recall that the largest clique in has size roughly . Let be
the supremum over such that there exists an algorithm that makes
queries in total to the adjacency matrix of , in a constant
number of rounds, and outputs a clique of size with
high probability. We give improved upper bounds on
for every and .
We also study analogous questions for finding subgraphs with density at least
for a given , and prove corresponding impossibility results.Comment: 19 pp, 5 figures, Focused Workshop on Networks and Their Limits held
at the Erd\H{o}s Center, Budapest, Hungary in July 202
Small rainbow cliques in randomly perturbed dense graphs
For two graphs and , write if has the property that every \emph{proper} colouring of its edges
yields a \emph{rainbow} copy of .
We study the thresholds for such so-called \emph{anti-Ramsey} properties in
randomly perturbed dense graphs, which are unions of the form , where is an -vertex graph with edge-density at least
, and is independent of .
In a companion article, we proved that the threshold for the property
is
, whenever . For smaller , the thresholds behave more erratically, and for
they deviate downwards significantly from the aforementioned
aesthetic form capturing the thresholds for \emph{large} cliques.
In particular, we show that the thresholds for are
, , and , respectively. For we
determine the threshold up to a -factor in the exponent: they are
and , respectively. For , the
threshold is ; this follows from a more general result about odd cycles
in our companion paper.Comment: 37 pages, several figures; update following reviewer(s) comment
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