260 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
Differential posets and restriction in critical groups
In recent work, Benkart, Klivans, and Reiner defined the critical group of a
faithful representation of a finite group , which is analogous to the
critical group of a graph. In this paper we study maps between critical groups
induced by injective group homomorphisms and in particular the map induced by
restriction of the representation to a subgroup. We show that in the abelian
group case the critical groups are isomorphic to the critical groups of a
certain Cayley graph and that the restriction map corresponds to a graph
covering map. We also show that when is an element in a differential tower
of groups, critical groups of certain representations are closely related to
words of up-down maps in the associated differential poset. We use this to
generalize an explicit formula for the critical group of the permutation
representation of the symmetric group given by the second author, and to
enumerate the factors in such critical groups.Comment: 18 pages; v2: minor edits and updated reference
Some open problems on permutation patterns
This is a brief survey of some open problems on permutation patterns, with an
emphasis on subjects not covered in the recent book by Kitaev, \emph{Patterns
in Permutations and words}. I first survey recent developments on the
enumeration and asymptotics of the pattern 1324, the last pattern of length 4
whose asymptotic growth is unknown, and related issues such as upper bounds for
the number of avoiders of any pattern of length for any given . Other
subjects treated are the M\"obius function, topological properties and other
algebraic aspects of the poset of permutations, ordered by containment, and
also the study of growth rates of permutation classes, which are containment
closed subsets of this poset.Comment: 20 pages. Related to upcoming talk at the British Combinatorial
Conference 2013. To appear in London Mathematical Society Lecture Note Serie
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