Eric Babson

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.

Cut sets and normed cohomology with applications to percolation

We discuss an inequality for graphs, which relates the distances between components of any minimal cut set to the lengths of generators for the homology of the graph. Our motivation arises from percolation theory. In particular this result is applied to Cayley graphs of finite presentations of groups with one end, where it gives an exponential bound on the number of minimal cut sets, and thereby shows that the critical probability for percolation on these graphs is neither zero nor one. We further show for this same class of graphs that the critical probability for the coalescence of all infinite components into a single one is neither zero nor one. Introduction Let G = (V,E) be a locally finite graph. Given two vertices u, v ∈ V , a (u, v) cut set Π ⊆ E is a set of edges that has nonempty intersection with every path from u to v. Similarly a (u,∞) cut set intersects every path from u contained in no finite subgraph. A (u, v) cut set is minimal (an mcs) if it contains no proper subset which is also a (u, v) cut set. Cut sets are naturally connected to homology. We will show how the distance between the two parts of any bipartition of a minimal cut set is related to the L∞ norm on the lattice of integral first cohomology classes and hence also to the L1 norm on homology. In particular we show that no bipartition of a minimal cut set can have its two parts separated by a distance of more than half the diameter of the first homology lattice. For Cayley graphs, this diameter is in turn related to the lengths of the relators. We present the inequality in the context of locally compact, complete, path metric spaces, associating to a graph the path metric in which the length of every edge is one. Our motivation comes from percolation theory [1], [3]. To date percolation has mainly been studied on Z, the Cayley graph for the free Abelian group with d free generators. The basic notion in percolation theory is pc(G), the critical probability for bond percolation on the graph G. In Bernoulli bond percolation the edges are independently colored open with probability p. Those edges that are not open are colored closed. The corresponding product measure on the edge colorings is denoted Pp,G or Pp. For a fixed open/closed edge coloring Eo ∪ Ec = E of a graph G with Received by the editors March 13, 1997. 1991 Mathematics Subject Classification. Primary 60K35. c ©1999 American Mathematical Society

Discrete Morse functions from lexicographic orders

This paper shows how to construct a discrete Morse function with a relatively small number of critical cells for the order complex of any finite poset with O and I from any lexicographic order on its maximal chains. Specifically, if we attach facets according to the lexicographic order on maximal chains, then each facet contributes at most one new face which is critical, and at most one Betti number changes; facets which do not change the homotopy type also do not contribute any critical faces. Dimensions of critical faces as well as a description of which facet attachments change the homotopy type are provided in terms of interval systems associated to the facets. As one application, the Mobius function may be computed as the alternating sum of Morse numbers. The above construction enables us to prove that the poset Π n /S λ of partitions of a set {1 λ 1,...,k λ k} with repetition is homotopy equivalent to a wedge of spheres of top dimension when A is a hook-shaped partition; it is likely that the proof may be extended to a larger class of A and perhaps to all A, despite a result of Ziegler (1986) which shows that Π n /S λ is not always Cohen-Macaulay.

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.

Cocircuit Graphs and Efficient Orientation Reconstruction in Oriented Matroids

We consider the cocircuit graph GMof an oriented matroid M, which is the 1-skeleton of the cell complex formed by the span of the cocircuits of M. As a result of Cordovil, Fukuda, and Guedes de Oliveira, the isomorphism class of M is not determined by GM, but it is determined if M is uniform and the vertices in GMare paired if they are associated to negative cocircuits; furthermore the reorientation class of an oriented matroid M with rank(M) ? 2 is determined by GMif every vertex in GMis labeled by the zero support of the associated cocircuit. In this paper we show that the isomorphism class of a uniform oriented matroid is determined by the cocircuit graph, and we present polynomial algorithms which provide constructive proofs to all these results. Furthermore it is shown that the correctness of the input of the algorithms can be verified in polynomial time.

The Permutations 123 p 4 … p m and 321 p 4 … p m are Wilf-Equivalent

Abstract. Write p1, p2…pm for the permutation matrix δpi, j. Let Sn (M) be the set of n×n permutation matrices which do not contain the m×m permutation matrix M as a submatrix. In [7] Simion and Schmidt show bijectively that |Sn (123) |=|Sn (213) |. In [9] this was generalised to a bijection between Sn (12 p3…pm) and Sn (21 p3…pm). In the present paper we obtain a bijection between Sn (123 p4…pm) and Sn (321 p4…pm).

Symmetric iterated Betti numbers

We define a set of invariants of a homogeneous ideal I in a polynomial ring called the symmetric iterated Betti numbers of I. We prove that for IΓ, the Stanley-Reisner ideal of a simplicial complex Γ, these numbers are the symmetric counterparts of the exterior iterated Betti numbers of Γ introduced by Dural and Rose, and that the extremal Betti numbers of IΓ are precisely the extremal (symmetric or exterior) iterated Betti numbers of Γ. We show that the symmetric iterated Betti numbers of an ideal I coincide with those of a particular reverse lexicographic generic initial ideal Gin(I) of I, and interpret these invariants in terms of the associated primes and standard pairs of Gin(I). We close with results and conjectures about the relationship between symmetric and exterior iterated Betti numbers of a simplicial complex.