Dmitry N. Kozlov

Complexes of graph homomorphisms

Hom(G, H) is a polyhedral complex defined for any two undirected graphsG andH. This construction was introduced by Lovász to give lower bounds for chromatic numbers of graphs. In this paper we initiate the study of the topological properties of this class of complexes.We prove that Hom(Km, Kn) is homotopy equivalent to a wedge of (n−m)-dimensional spheres, and provide an enumeration formula for the number of the spheres. As a corollary we prove that if for some graphG, and integersm≥2 andk≥−1, we have ϖ1k(Hom(Km, G))≠0, thenχ(G)≥k+m; here ℤ2-action is induced by the swapping of two vertices inKm, and ϖ1 is the first Stiefel-Whitney class corresponding to this action.Furthermore, we prove that a fold in the first argument of Hom(G, H) induces a homotopy equivalence. It then follows that Hom(F, Kn) is homotopy equivalent to a direct product of (n−2)-dimensional spheres, while Hom(F, Kn) is homotopy equivalent to a wedge of spheres, whereF is an arbitrary forest andF is its complement.

Complexes of Directed Trees

To every directed graph G one can associate a complex ?(G) consisting of directed subforests. This construction, suggested to us by R. Stanley, is especially important in the case of a complete double directed graph Gn, where it leads to the study of some interesting representations of the symmetric group and corresponds (via the Stanley?Reisner correspondence) to an interesting quotient ring. Our main result states that ?(Gn) is shellable, in particular, Cohen?Macaulay, which can be further translated to say that the Stanley?Reisner ring of ?(Gn) is Cohen?Macaulay. Besides that, by computing the homology groups of ?(G) for the cases when G is essentially a tree and when G is a double directed cycle, we touch upon the general question of the interaction of the combinatorial properties of a graph and the topological properties of the associated complex.

General lexicographic shellability and orbit arrangements

We introduce a new poset property which we call EC-shellability. It is more general than the more established concept of EL-shellability, but it still implies shellability. Because of Theorem 3.10, EC-shellability is entitled to be called general lexicographic shellability.As an application of our new concept, we prove that intersection lattices Πλ of orbit arrangementsAλ are EC-shellable for a very large class of partitions λ. This allows us to compute the topology of the link and the complement for these arrangements. In particular, for this class of λs, we are able to settle a conjecture of Björner [B94, Conjecture 13.3.2], stating that the cohomology groups of the complement of the orbit arrangements are torsion-free.We also present a class of partitions for which Πλ is not shellable, along with other issues scattered throughout the paper.

Topological obstructions to graph colorings

For any two graphs G and H Lovasz has defined a cell complex Hom (G,H) having in mind the general program that the algebraic invariants of these complexes should provide obstructions to graph colorings. Here we announce the proof of a conjecture of Lovasz concerning these complexes with G a cycle of odd length. More specifically, we show that If Hom (C2r+1, G) is k-connected, then χ(G) ≥ k + 4. Our actual statement is somewhat sharper, as we find obstructions already in the nonvanishing of powers of certain Stiefel-Whitney classes.

The threshold function for vanishing of the top homology group of random

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The geometry of coin-weighing problems

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Simple homotopy types of Hom-complexes, neighborhood complexes, Lovász complexes, and atom crosscut complexes☆

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Homotopy Type of Complexes of Graph Homomorphisms between Cycles

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