2,867 research outputs found
A sharp threshold for random graphs with a monochromatic triangle in every edge coloring
Let be the set of all finite graphs with the Ramsey property that
every coloring of the edges of by two colors yields a monochromatic
triangle. In this paper we establish a sharp threshold for random graphs with
this property. Let be the random graph on vertices with edge
probability . We prove that there exists a function with
, as tends to infinity
Pr[G(n,(1-\eps)\hat c/\sqrt{n}) \in \R ] \to 0 and Pr [ G(n,(1+\eps)\hat
c/\sqrt{n}) \in \R ] \to 1. A crucial tool that is used in the proof and is
of independent interest is a generalization of Szemer\'edi's Regularity Lemma
to a certain hypergraph setting.Comment: 101 pages, Final version - to appear in Memoirs of the A.M.
Lower bounds for Max-Cut in -free graphs via semidefinite programming
For a graph , let denote the size of the maximum cut in . The
problem of estimating as a function of the number of vertices and edges
of has a long history and was extensively studied in the last fifty years.
In this paper we propose an approach, based on semidefinite programming (SDP),
to prove lower bounds on . We use this approach to find large cuts in
graphs with few triangles and in -free graphs.Comment: 21 pages, to be published in LATIN 2020 proceedings, Updated version
is rewritten to include additional results along with corrections to original
argument
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
Triangle-free subgraphs of random graphs
Recently there has been much interest in studying random graph analogues of
well known classical results in extremal graph theory. Here we follow this
trend and investigate the structure of triangle-free subgraphs of with
high minimum degree. We prove that asymptotically almost surely each
triangle-free spanning subgraph of with minimum degree at least
is -close to bipartite,
and each spanning triangle-free subgraph of with minimum degree at
least is -close to
-partite for some . These are random graph analogues of a
result by Andr\'asfai, Erd\H{o}s, and S\'os [Discrete Math. 8 (1974), 205-218],
and a result by Thomassen [Combinatorica 22 (2002), 591--596]. We also show
that our results are best possible up to a constant factor.Comment: 18 page
Letter graphs and geometric grid classes of permutations: characterization and recognition
In this paper, we reveal an intriguing relationship between two seemingly
unrelated notions: letter graphs and geometric grid classes of permutations. An
important property common for both of them is well-quasi-orderability,
implying, in a non-constructive way, a polynomial-time recognition of geometric
grid classes of permutations and -letter graphs for a fixed . However,
constructive algorithms are available only for . In this paper, we present
the first constructive polynomial-time algorithm for the recognition of
-letter graphs. It is based on a structural characterization of graphs in
this class.Comment: arXiv admin note: text overlap with arXiv:1108.6319 by other author
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