19 research outputs found
A new bound for the 2/3 conjecture
We show that any n-vertex complete graph with edges colored with three colors
contains a set of at most four vertices such that the number of the neighbors
of these vertices in one of the colors is at least 2n/3. The previous best
value, proved by Erdos, Faudree, Gould, Gy\'arf\'as, Rousseau and Schelp in
1989, is 22. It is conjectured that three vertices suffice
A solution to the 2/3 conjecture
We prove a vertex domination conjecture of Erd\H os, Faudree, Gould,
Gy\'arf\'as, Rousseau, and Schelp, that for every n-vertex complete graph with
edges coloured using three colours there exists a set of at most three vertices
which have at least 2n/3 neighbours in one of the colours. Our proof makes
extensive use of the ideas presented in "A New Bound for the 2/3 Conjecture" by
Kr\'al', Liu, Sereni, Whalen, and Yilma.Comment: 12 pages, 4 figures, 2 data files and proof checking code. Revised
version to appear in SIAM Journal on Discrete Mathematic
Counting flags in triangle-free digraphs
Motivated by the Caccetta-Haggkvist Conjecture, we prove that every digraph
on n vertices with minimum outdegree 0.3465n contains an oriented triangle.
This improves the bound of 0.3532n of Hamburger, Haxell and Kostochka. The main
new tool we use in our proof is the theory of flag algebras developed recently
by Razborov.Comment: 19 pages, 7 figures; this is the final version to appear in
Combinatoric
Finitely forcible graphons and permutons
We investigate when limits of graphs (graphons) and permutations (permutons)
are uniquely determined by finitely many densities of their substructures,
i.e., when they are finitely forcible. Every permuton can be associated with a
graphon through the notion of permutation graphs. We find permutons that are
finitely forcible but the associated graphons are not. We also show that all
permutons that can be expressed as a finite combination of monotone permutons
and quasirandom permutons are finitely forcible, which is the permuton
counterpart of the result of Lovasz and Sos for graphons.Comment: 30 pages, 18 figure
Compactness and finite forcibility of graphons
Graphons are analytic objects associated with convergent sequences of graphs.
Problems from extremal combinatorics and theoretical computer science led to a
study of graphons determined by finitely many subgraph densities, which are
referred to as finitely forcible. Following the intuition that such graphons
should have finitary structure, Lovasz and Szegedy conjectured that the
topological space of typical vertices of a finitely forcible graphon is always
compact. We disprove the conjecture by constructing a finitely forcible graphon
such that the associated space is not compact. The construction method gives a
general framework for constructing finitely forcible graphons with non-trivial
properties
Finitely forcible graphons with an almost arbitrary structure
Graphons are analytic objects representing convergent sequences of large
graphs. A graphon is said to be finitely forcible if it is determined by
finitely many subgraph densities, i.e., if the asymptotic structure of graphs
represented by such a graphon depends only on finitely many density
constraints. Such graphons appear in various scenarios, particularly in
extremal combinatorics.
Lovasz and Szegedy conjectured that all finitely forcible graphons possess a
simple structure. This was disproved in a strong sense by Cooper, Kral and
Martins, who showed that any graphon is a subgraphon of a finitely forcible
graphon. We strenghten this result by showing for every that
any graphon spans a proportion of a finitely forcible graphon
Finitely forcible graphons with an almost arbitrary structure
Graphons are analytic objects representing convergent sequences of large graphs. A graphon is said to be finitely forcible if it is determined by finitely many subgraph densities, i.e., if the asymptotic structure of graphs represented by such a graphon depends only on finitely many density constraints. Such graphons appear in various scenarios, particularly in extremal combinatorics.
Lovasz and Szegedy conjectured that all finitely forcible graphons possess a simple structure. This was disproved in a strong sense by Cooper, Kral and Martins, who showed that any graphon is a subgraphon of a finitely forcible graphon. We strenghten this result by showing for every Îľ>0 that any graphon spans a 1âÎľ proportion of a finitely forcible graphon