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Topology of Hom complexes and test graphs for bounding chromatic number

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 Added by Anton Dochtermann
 Publication date 2009
  fields
and research's language is English




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We introduce new methods for understanding the topology of $Hom$ complexes (spaces of homomorphisms between two graphs), mostly in the context of group actions on graphs and posets. We view $Hom(T,-)$ and $Hom(-,G)$ as functors from graphs to posets, and introduce a functor $(-)^1$ from posets to graphs obtained by taking atoms as vertices. Our main structural results establish useful interpretations of the equivariant homotopy type of $Hom$ complexes in terms of spaces of equivariant poset maps and $Gamma$-twisted products of spaces. When $P = F(X)$ is the face poset of a simplicial complex $X$, this provides a useful way to control the topology of $Hom$ complexes. Our foremost application of these results is the construction of new families of `test graphs with arbitrarily large chromatic number - graphs $T$ with the property that the connectivity of $Hom(T,G)$ provides the best possible lower bound on the chromatic number of $G$. In particular we focus on two infinite families, which we view as higher dimensional analogues of odd cycles. The family of `spherical graphs have connections to the notion of homomorphism duality, whereas the family of `twisted toroidal graphs lead us to establish a weakened version of a conjecture (due to Lov{a}sz) relating topological lower bounds on chromatic number to maximum degree. Other structural results allow us to show that any finite simplicial complex $X$ with a free action by the symmetric group $S_n$ can be approximated up to $S_n$-homotopy equivalence as $Hom(K_n,G)$ for some graph $G$; this is a generalization of a result of Csorba. We conclude the paper with some discussion regarding the underlying categorical notions involved in our study.



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99 - Anton Dochtermann 2006
We investigate a notion of $times$-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 $times$-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 $times$-homotopy naturally leads to a notion of homotopy equivalence which we show has several equivalent characterizations. We apply the notions of $times$-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 `$A$-theory associated to the cartesian product in the category of reflexive graphs.
87 - Anton Dochtermann 2007
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