293 research outputs found
Hom complexes and homotopy theory in the category of graphs
We investigate a notion of -homotopy of graph maps that is based on
the internal hom associated to the categorical product in the category of
graphs. It is shown that graph -homotopy is characterized by the
topological properties of the \Hom complex, a functorial way to assign a
poset (and hence topological space) to a pair of graphs; \Hom complexes were
introduced by Lov\'{a}sz and further studied by Babson and Kozlov to give
topological bounds on chromatic number. Along the way, we also establish some
structural properties of \Hom complexes involving products and exponentials
of graphs, as well as a symmetry result which can be used to reprove a theorem
of Kozlov involving foldings of graphs. Graph -homotopy naturally leads
to a notion of homotopy equivalence which we show has several equivalent
characterizations. We apply the notions of -homotopy equivalence to the
class of dismantlable graphs to get a list of conditions that again
characterize these. We end with a discussion of graph homotopies arising from
other internal homs, including the construction of `-theory' associated to
the cartesian product in the category of reflexive graphs.Comment: 28 pages, 13 figures, final version, to be published in European J.
Com
Homotopy groups of Hom complexes of graphs
The notion of -homotopy from \cite{DocHom} is investigated in the
context of the category of pointed graphs. The main result is a long exact
sequence that relates the higher homotopy groups of the space \Hom_*(G,H)
with the homotopy groups of \Hom_*(G,H^I). Here \Hom_*(G,H) is a space
which parametrizes pointed graph maps from to (a pointed version of the
usual \Hom complex), and is the graph of based paths in . As a
corollary it is shown that \pi_i \big(\Hom_*(G,H) \big) \cong [G,\Omega^i
H]_{\times}, where is the graph of based closed paths in and
is the set of -homotopy classes of pointed graph maps
from to . This is similar in spirit to the results of \cite{BBLL}, where
the authors seek a space whose homotopy groups encode a similarly defined
homotopy theory for graphs. The categorical connections to those constructions
are discussed.Comment: 20 pages, 6 figures, final version, to be published in J. Combin.
Theory Ser.
Tropical types and associated cellular resolutions
An arrangement of finitely many tropical hyperplanes in the tropical torus
leads to a notion of `type' data for points, with the underlying unlabeled
arrangement giving rise to `coarse type'. It is shown that the decomposition of
the tropical torus induced by types gives rise to minimal cocellular
resolutions of certain associated monomial ideals. Via the Cayley trick from
geometric combinatorics this also yields cellular resolutions supported on
mixed subdivisions of dilated simplices, extending previously known
constructions. Moreover, the methods developed lead to an algebraic algorithm
for computing the facial structure of arbitrary tropical complexes from point
data.Comment: minor correction
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