30 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.
Homomorphism complexes, reconfiguration, and homotopy for directed graphs
The neighborhood complex of a graph was introduced by Lov\'asz to provide
topological lower bounds on chromatic number. More general homomorphism
complexes of graphs were further studied by Babson and Kozlov. Such `Hom
complexes' are also related to mixings of graph colorings and other
reconfiguration problems, as well as a notion of discrete homotopy for graphs.
Here we initiate the detailed study of Hom complexes for directed graphs
(digraphs). For any pair of digraphs graphs and , we consider the
polyhedral complex that parametrizes the directed graph
homomorphisms . Hom complexes of digraphs have applications
in the study of chains in graded posets and cellular resolutions of monomial
ideals. We study examples of directed Hom complexes and relate their
topological properties to certain graph operations including products,
adjunctions, and foldings. We introduce a notion of a neighborhood complex for
a digraph and prove that its homotopy type is recovered as the Hom complex of
homomorphisms from a directed edge. We establish a number of results regarding
the topology of directed neighborhood complexes, including the dependence on
directed bipartite subgraphs, a digraph version of the Mycielski construction,
as well as vanishing theorems for higher homology. The Hom complexes of
digraphs provide a natural framework for reconfiguration of homomorphisms of
digraphs. Inspired by notions of directed graph colorings we study the
connectivity of for a tournament. Finally, we use
paths in the internal hom objects of digraphs to define various notions of
homotopy, and discuss connections to the topology of Hom complexes.Comment: 34 pages, 10 figures; V2: some changes in notation, clarified
statements and proofs, other corrections and minor revisions incorporating
comments from referee