1,516 research outputs found
Ramsey expansions of metrically homogeneous graphs
We discuss the Ramsey property, the existence of a stationary independence
relation and the coherent extension property for partial isometries (coherent
EPPA) for all classes of metrically homogeneous graphs from Cherlin's
catalogue, which is conjectured to include all such structures. We show that,
with the exception of tree-like graphs, all metric spaces in the catalogue have
precompact Ramsey expansions (or lifts) with the expansion property. With two
exceptions we can also characterise the existence of a stationary independence
relation and the coherent EPPA.
Our results can be seen as a new contribution to Ne\v{s}et\v{r}il's
classification programme of Ramsey classes and as empirical evidence of the
recent convergence in techniques employed to establish the Ramsey property, the
expansion (or lift or ordering) property, EPPA and the existence of a
stationary independence relation. At the heart of our proof is a canonical way
of completing edge-labelled graphs to metric spaces in Cherlin's classes. The
existence of such a "completion algorithm" then allows us to apply several
strong results in the areas that imply EPPA and respectively the Ramsey
property.
The main results have numerous corollaries on the automorphism groups of the
Fra\"iss\'e limits of the classes, such as amenability, unique ergodicity,
existence of universal minimal flows, ample generics, small index property,
21-Bergman property and Serre's property (FA).Comment: 57 pages, 14 figures. Extends results of arXiv:1706.00295. Minor
revisio
Super edge-magic deficiency of join-product graphs
A graph is called \textit{super edge-magic} if there exists a bijective
function from to such
that and is a
constant for every edge of . Furthermore, the \textit{super
edge-magic deficiency} of a graph is either the minimum nonnegative integer
such that is super edge-magic or if there exists no
such integer.
\emph{Join product} of two graphs is their graph union with additional edges
that connect all vertices of the first graph to each vertex of the second
graph. In this paper, we study the super edge-magic deficiencies of a wheel
minus an edge and join products of a path, a star, and a cycle, respectively,
with isolated vertices.Comment: 11 page
Langford sequences and a product of digraphs
Skolem and Langford sequences and their many generalizations have
applications in numerous areas. The -product is a generalization of
the direct product of digraphs. In this paper we use the -product
and super edge-magic digraphs to construct an exponential number of Langford
sequences with certain order and defect. We also apply this procedure to
extended Skolem sequences.Comment: 10 pages, 6 figures, to appear in European Journal of Combinatoric
Quantum automorphism groups of homogeneous graphs
Associated to a finite graph is its quantum automorphism group . The
main problem is to compute the Poincar\'e series of , meaning the series
whose coefficients are multiplicities of 1 into tensor
powers of the fundamental representation. In this paper we find a duality
between certain quantum groups and planar algebras, which leads to a planar
algebra formulation of the problem. Together with some other results, this
gives for all homogeneous graphs having 8 vertices or less.Comment: 30 page
Ehrhart Series of Polytopes Related to Symmetric Doubly-Stochastic Matrices
In Ehrhart theory, the -vector of a rational polytope often provide
insights into properties of the polytope that may be otherwise obscured. As an
example, the Birkhoff polytope, also known as the polytope of real
doubly-stochastic matrices, has a unimodal -vector, but when even small
modifications are made to the polytope, the same property can be very difficult
to prove. In this paper, we examine the -vectors of a class of polytopes
containing real doubly-stochastic symmetric matrices.Comment: 11 pages; this revision removes an erroneous proposition from earlier
versions and expands on the implication
Group twin coloring of graphs
For a given graph , the least integer such that for every
Abelian group of order there exists a proper edge labeling
so that for each edge is called the \textit{group twin
chromatic index} of and denoted by . This graph invariant is
related to a few well-known problems in the field of neighbor distinguishing
graph colorings. We conjecture that for all graphs
without isolated edges, where is the maximum degree of , and
provide an infinite family of connected graph (trees) for which the equality
holds. We prove that this conjecture is valid for all trees, and then apply
this result as the base case for proving a general upper bound for all graphs
without isolated edges: , where
denotes the coloring number of . This improves the best known
upper bound known previously only for the case of cyclic groups
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