1,077,239 research outputs found
Generalised morphisms of k-graphs: k-morphs
In a number of recent papers, (k+l)-graphs have been constructed from
k-graphs by inserting new edges in the last l dimensions. These constructions
have been motivated by C*-algebraic considerations, so they have not been
treated systematically at the level of higher-rank graphs themselves. Here we
introduce k-morphs, which provide a systematic unifying framework for these
various constructions. We think of k-morphs as the analogue, at the level of
k-graphs, of C*-correspondences between C*-algebras. To make this analogy
explicit, we introduce a category whose objects are k-graphs and whose
morphisms are isomorphism classes of k-morphs. We show how to extend the
assignment \Lambda \mapsto C*(\Lambda) to a functor from this category to the
category whose objects are C*-algebras and whose morphisms are isomorphism
classes of C*-correspondences.Comment: 27 pages, four pictures drawn with Tikz. Version 2: title changed and
numerous minor corrections and improvements. This version to appear in Trans.
Amer. Math. So
Packing k-partite k-uniform hypergraphs
Let and be -graphs (-uniform hypergraphs); then a perfect
-packing in is a collection of vertex-disjoint copies of in
which together cover every vertex of . For any fixed let
be the minimum such that any -graph on vertices with
minimum codegree contains a perfect -packing. The
problem of determining has been widely studied for graphs (i.e.
-graphs), but little is known for . Here we determine the
asymptotic value of for all complete -partite -graphs ,
as well as a wide class of other -partite -graphs. In particular, these
results provide an asymptotic solution to a question of R\"odl and Ruci\'nski
on the value of when is a loose cycle. We also determine
asymptotically the codegree threshold needed to guarantee an -packing
covering all but a constant number of vertices of for any complete
-partite -graph .Comment: v2: Updated with minor corrections. Accepted for publication in
Journal of Combinatorial Theory, Series
The degree-diameter problem for sparse graph classes
The degree-diameter problem asks for the maximum number of vertices in a
graph with maximum degree and diameter . For fixed , the answer
is . We consider the degree-diameter problem for particular
classes of sparse graphs, and establish the following results. For graphs of
bounded average degree the answer is , and for graphs of
bounded arboricity the answer is \Theta(\Delta^{\floor{k/2}}), in both cases
for fixed . For graphs of given treewidth, we determine the the maximum
number of vertices up to a constant factor. More precise bounds are given for
graphs of given treewidth, graphs embeddable on a given surface, and
apex-minor-free graphs
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
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