We apply some basic notions from combinatorial topology to establish various algebraic properties of edge ideals of graphs and more general Stanley-Reisner rings. In this way we provide new short proofs of some theorems from the literature regarding linearity, Betti numbers, and (sequentially) Cohen-Macaulay properties of edges ideals associated to chordal, complements of chordal, and Ferrers graphs, as well as trees and forests. Our approach unifies (and in many cases strengthens) these results and also provides combinatorial/enumerative interpretations of certain algebraic properties. We apply our setup to obtain new results regarding algebraic properties of edge ideals in the context of local changes to a graph (adding whiskers and ears) as well as bounded vertex degree. These methods also lead to recursive relations among certain generating functions of Betti numbers which we use to establish new formulas for the projective dimension of edge ideals. We use only well-known tools from combinatorial topology along the lines of independence complexes of graphs, (not necessarily pure) vertex decomposability, shellability, etc.
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 Lovász 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 'A-theory' associated to the cartesian product in the category of reflexive graphs. * Research supported in part by NSF grant DMS-9983797 of (compactly generated) topological spaces, where a homotopy between maps f : X → Y and g : X → Y is nothing more than a map from the interval I into the topological space Map(X, Y ).Other examples include simplicial objects, as well as the category of chain complexes of R-modules.For the latter, a chain homotopy between chain maps f : C → D and g : C → D can be recovered as a map from the chain complex I (defined to be the complex consisting of 0 in all dimensions except R in dimensions 0 and 1, with the identity map between them) into the complex Hom(C, D).In this paper we consider these constructions in the context of the category of graphs. In particular, we investigate a notion of what we call ×-homotopy that arises from consideration of the well known internal hom associated to the categorical product. Here the relevant construction is the exponential H G , a graph whose looped vertices parameterize the graph homomorphisms (maps) from G to H. We use the notion of (graph theoretic) connectivity to provide a notion of a 'path' in the exponential graph. It turns out that ×-homotopy classes of maps are related to the topology of the so-called Hom-complex, a functorial way to assign a poset Hom(G, H) (and hence topological space) to a pair of graphs G and H. Hom complexes were first introduced by Lovàsz in his celebrated proof of the Kneser conjecture (see [17]), and were later developed by Babson and Kozlov in their proof of the Lovàsz conjecture (see [2] and [3]). Elements of the poset Hom(G, H) are graph multi-homomorphisms from G to H, with the set of graph homomorphisms the atoms. Fixing one of the two coordinates of the Hom complex in each case provides a functor from graphs to topological spaces, and in Theorem 5.1 we show that ×-homotopy of graph maps can be characterized by the topological homotopy type of the maps...
The Hom complex of homomorphisms between two graphs was originally introduced to provide topological lower bounds on chromatic number. In this paper we introduce new methods for understanding the topology of Hom complexes, mostly in the context of Γ-actions on graphs and posets (for some group Γ). We view the Hom(T, •) and Hom(•, G) complexes 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 Γ-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. These constructions generalize those of the second author from [17] as well as the calculation of the homotopy groups of Hom complexes from [8].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 Lovász) 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 Sn can be approximated up to Sn-homotopy equivalence as Hom(Kn, G) for some graph G; this is a generalization of the results of Csorba from [5] for the case of n = 2. We conclude the paper with some discussion regarding the underlying categorical notions involved in our study.
Minimal cellular resolutions of the edge ideals of cointerval hypergraphs are constructed. This class of d-uniform hypergraphs coincides with the complements of interval graphs (for the case d = 2), and strictly contains the class of 'strongly stable' hypergraphs corresponding to pure shifted simplicial complexes. The polyhedral complexes supporting the resolutions are described as certain spaces of directed graph homomorphisms, and are realized as subcomplexes of mixed subdivisions of the Minkowski sums of simplices. Resolutions of more general hypergraphs are obtained by considering decompositions into cointerval hypergraphs.
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