440 research outputs found
Chromatic roots and limits of dense graphs
In this short note we observe that recent results of Abert and Hubai and of
Csikvari and Frenkel about Benjamini--Schramm continuity of the holomorphic
moments of the roots of the chromatic polynomial extend to the theory of dense
graph sequences. We offer a number of problems and conjectures motivated by
this observation.Comment: 9 page
Cut distance identifying graphon parameters over weak* limits
The theory of graphons comes with the so-called cut norm and the derived cut
distance. The cut norm is finer than the weak* topology. Dole\v{z}al and
Hladk\'y [Cut-norm and entropy minimization over weak* limits, J. Combin.
Theory Ser. B 137 (2019), 232-263] showed, that given a sequence of graphons, a
cut distance accumulation graphon can be pinpointed in the set of weak*
accumulation points as a minimizer of the entropy. Motivated by this, we study
graphon parameters with the property that their minimizers or maximizers
identify cut distance accumulation points over the set of weak* accumulation
points. We call such parameters cut distance identifying. Of particular
importance are cut distance identifying parameters coming from subgraph
densities, t(H,*). This concept is closely related to the emerging field of
graph norms, and the notions of the step Sidorenko property and the step
forcing property introduced by Kr\'al, Martins, Pach and Wrochna [The step
Sidorenko property and non-norming edge-transitive graphs, J. Combin. Theory
Ser. A 162 (2019), 34-54]. We prove that a connected graph is weakly norming if
and only if it is step Sidorenko, and that if a graph is norming then it is
step forcing. Further, we study convexity properties of cut distance
identifying graphon parameters, and find a way to identify cut distance limits
using spectra of graphons. We also show that continuous cut distance
identifying graphon parameters have the "pumping property", and thus can be
used in the proof of the the Frieze-Kannan regularity lemma.Comment: 48 pages, 3 figures. Correction when treating disconnected norming
graphs, and a new section 3.2 on index pumping in the regularity lemm
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 ε>0 that any graphon spans a 1−ε proportion of a finitely forcible graphon
Sidorenko's conjecture, colorings and independent sets
Let denote the number of homomorphisms from a graph to a
graph . Sidorenko's conjecture asserts that for any bipartite graph , and
a graph we have where
and denote the number of vertices and edges of the graph and
, respectively. In this paper we prove Sidorenko's conjecture for certain
special graphs : for the complete graph on vertices, for a
with a loop added at one of the end vertices, and for a path on vertices
with a loop added at each vertex. These cases correspond to counting colorings,
independent sets and Widom-Rowlinson colorings of a graph . For instance,
for a bipartite graph the number of -colorings
satisfies
In fact, we will prove that in the last two cases (independent sets and
Widom-Rowlinson colorings) the graph does not need to be bipartite. In all
cases, we first prove a certain correlation inequality which implies
Sidorenko's conjecture in a stronger form.Comment: Two references added and Remark 2.1 is expande
The step Sidorenko property and non-norming edge-transitive graphs
Sidorenko's Conjecture asserts that every bipartite graph H has the Sidorenko
property, i.e., a quasirandom graph minimizes the density of H among all graphs
with the same edge density. We study a stronger property, which requires that a
quasirandom multipartite graph minimizes the density of H among all graphs with
the same edge densities between its parts; this property is called the step
Sidorenko property. We show that many bipartite graphs fail to have the step
Sidorenko property and use our results to show the existence of a bipartite
edge-transitive graph that is not weakly norming; this answers a question of
Hatami [Israel J. Math. 175 (2010), 125-150].Comment: Minor correction on page
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