104,471 research outputs found
k --Universal Finite Graphs
This paper investigates the class of k-universal finite graphs, a local
analog of the class of universal graphs, which arises naturally in the study of
finite variable logics. The main results of the paper, which are due to Shelah,
establish that the class of k-universal graphs is not definable by an infinite
disjunction of first-order existential sentences with a finite number of
variables and that there exist k-universal graphs with no k-extendible induced
subgraphs
Universal targets for homomorphisms of edge-colored graphs
A -edge-colored graph is a finite, simple graph with edges labeled by
numbers . A function from the vertex set of one -edge-colored
graph to another is a homomorphism if the endpoints of any edge are mapped to
two different vertices connected by an edge of the same color. Given a class
of graphs, a -edge-colored graph (not necessarily
with the underlying graph in ) is -universal for
when any -edge-colored graph with the underlying graph in
admits a homomorphism to . We characterize graph classes that admit
-universal graphs. For such classes, we establish asymptotically almost
tight bounds on the size of the smallest universal graph.
For a nonempty graph , the density of is the maximum ratio of the
number of edges to the number of vertices ranging over all nonempty subgraphs
of . For a nonempty class of graphs, denotes
the density of , that is the supremum of densities of graphs in
.
The main results are the following. The class admits
-universal graphs for if and only if there is an absolute constant
that bounds the acyclic chromatic number of any graph in . For any
such class, there exists a constant , such that for any , the size
of the smallest -universal graph is between and
.
A connection between the acyclic coloring and the existence of universal
graphs was first observed by Alon and Marshall (Journal of Algebraic
Combinatorics, 8(1):5-13, 1998). One of their results is that for planar
graphs, the size of the smallest -universal graph is between and
. Our results yield that there exists a constant such that for all
, this size is bounded from above by
Big Ramsey degrees in universal inverse limit structures
We build a collection of topological Ramsey spaces of trees giving rise to
universal inverse limit structures, extending Zheng's work for the profinite
graph to the setting of Fra\"{\i}ss\'{e} classes of finite ordered binary
relational structures with the Ramsey property. This work is based on the
Halpern-L\"{a}uchli theorem, but different from the Milliken space of strong
subtrees. Based on these topological Ramsey spaces and the work of
Huber-Geschke-Kojman on inverse limits of finite ordered graphs, we prove that
for each such Fra\"{\i}ss\'{e} class, its universal inverse limit structures
has finite big Ramsey degrees under finite Baire-measurable colourings. For
finite ordered graphs, finite ordered -clique free graphs (),
finite ordered oriented graphs, and finite ordered tournaments, we characterize
the exact big Ramsey degrees.Comment: 20 pages, 5 figure
Infinitesimal deformations of Poisson bi-vectors using the Kontsevich graph calculus
Let be a Poisson structure on a finite-dimensional affine real manifold.
Can be deformed in such a way that it stays Poisson? The language of
Kontsevich graphs provides a universal approach -- with respect to all affine
Poisson manifolds -- to finding a class of solutions to this deformation
problem. For that reasoning, several types of graphs are needed. In this paper
we outline the algorithms to generate those graphs. The graphs that encode
deformations are classified by the number of internal vertices ; for we present all solutions of the deformation problem. For , first reproducing the pentagon-wheel picture suggested at
by Kontsevich and Willwacher, we construct the heptagon-wheel cocycle that
yields a new unique solution without -loops and tadpoles at .Comment: International conference ISQS'25 on integrable systems and quantum
symmetries (6-10 June 2017 in CVUT Prague, Czech Republic). Introductory
paragraph I.1 follows p.3 in arXiv:1710.00658 [math.CO]; 13 pages, 3 figures,
2 table
The Patterson-Sullivan embedding and minimal volume entropy for outer space
Motivated by Bonahon's result for hyperbolic surfaces, we construct an
analogue of the Patterson-Sullivan-Bowen-Margulis map from the Culler-Vogtmann
outer space into the space of projectivized geodesic currents on a
free group. We prove that this map is a topological embedding. We also prove
that for every the minimum of the volume entropy of the universal
covers of finite connected volume-one metric graphs with fundamental group of
rank and without degree-one vertices is equal to and that
this minimum is realized by trivalent graphs with all edges of equal lengths,
and only by such graphs.Comment: An updated versio
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