Russ Woodroofe

Vertex decomposable graphs and obstructions to shellability

Inspired by several recent papers on the edge ideal of a graph G, we study the equivalent notion of the independence complex of G. Using the tool of vertex decomposability from geometric combinatorics, we show that 5-chordal graphs with no chordless 4-cycles are shellable and sequentially Cohen-Macaulay. We use this result to characterize the obstructions to shellability in flag complexes, extending work of Billera, Myers, and Wachs. We also show how vertex decomposability may be used to show that certain graph constructions preserve shellability.

Matchings, coverings, and Castelnuovo-Mumford regularity

We show that the co-chordal cover number of a graph G gives an upper bound for the Castelnuovo-Mumford regularity of the associated edge ideal. Several known combinatorial upper bounds of regularity for edge ideals are then easy consequences of covering results from graph theory, and we derive new upper bounds by looking at additional covering results.

The geometric maximum traveling salesman problem

We consider the traveling salesman problem when the cities are points in &##x211D;^<i>d</i> for some fixed <i>d</i> and distances are computed according to geometric distances, determined by some norm. We show that for any polyhedral norm, the problem of finding a tour of <i>maximum</i> length can be solved in polynomial time. If arithmetic operations are assumed to take unit time, our algorithms run in time <i>O</i>(<i>n</i>^<i>f</i>−2 log <i>n</i>), where <i>f</i> is the number of facets of the polyhedron determining the polyhedral norm. Thus, for example, we have <i>O</i>(<i>n</i>² log <i>n</i>) algorithms for the cases of points in the plane under the Rectilinear and Sup norms. This is in contrast to the fact that finding a <i>minimum</i> length tour in each case is NP-hard. Our approach can be extended to the more general case of <i>quasi-norms</i> with a not necessarily symmetric unit ball, where we get a complexity of <i>O</i>(<i>n</i>^{2<i>f</i>−2} log <i>n</i>).For the special case of two-dimensional metrics with <i>f</i> = 4 (which includes the Rectilinear and Sup norms), we present a simple algorithm with <i>O</i>(<i>n</i>) running time. The algorithm does not use any indirect addressing, so its running time remains valid even in comparison based models in which sorting requires Ω(<i>n</i> log <i>n</i>) time. The basic mechanism of the algorithm provides some intuition on why polyhedral norms allow fast algorithms.Complementing the results on simplicity for polyhedral norms, we prove that, for the case of Euclidean distances in &##x211D;^<i>d</i> for <i>d</i> ≥ 3, the Maximum TSP is NP-hard. This sheds new light on the well-studied difficulties of Euclidean distances.

RESULTS ON THE REGULARITY OF SQUARE-FREE MONOMIAL IDEALS

In a 2008 paper, the first author and Van Tuyl proved that the regularity of the edge ideal of a graph G is at most one greater than the matching number of G. In this note, we provide a generalization of this result to any square-free monomial ideal. We define a 2-collage in a simple hypergraph to be a collection of edges with the property that for any edge E of the hypergraph, there exists an edge F in the collage such that |E \ F| < 2. The Castelnuovo-Mumford regularity of the edge ideal of a simple hypergraph is bounded above by a multiple of the minimum size of a 2-collage. We also give a recursive formula to compute the regularity of a vertex-decomposable hypergraph. Finally, we show that regularity in the graph case is bounded by a certain statistic based on maximal packings of nondegenerate star subgraphs.

Erdős-Ko-Rado theorems for simplicial complexes

A recent framework for generalizing the Erdos-Ko-Rado theorem, due to Holroyd, Spencer, and Talbot, defines the Erdos-Ko-Rado property for a graph in terms of the graphs independent sets. Since the family of all independent sets of a graph forms a simplicial complex, it is natural to further generalize the Erdos-Ko-Rado property to an arbitrary simplicial complex. An advantage of working in simplicial complexes is the availability of algebraic shifting, a powerful shifting (compression) technique, which we use to verify a conjecture of Holroyd and Talbot in the case of sequentially Cohen-Macaulay near-cones.

A new subgroup lattice characterization of finite solvable groups

Abstract We show that if G is a finite group then no chain of modular elements in its subgroup lattice L ( G ) is longer than a chief series. Also, we show that if G is a nonsolvable finite group then every maximal chain in L ( G ) has length at least two more than the chief length of G , thereby providing a converse of a result of J. Kohler. Our results enable us to give a new characterization of finite solvable groups involving only the combinatorics of subgroup lattices. Namely, a finite group G is solvable if and only if L ( G ) contains a maximal chain X and a chain M consisting entirely of modular elements, such that X and M have the same length.

Order complexes of coset posets of finite groups are not contractible

Abstract We show that the order complex of the poset of all cosets of all proper subgroups of a finite group G is never F 2 -acyclic and therefore never contractible. This settles a question of K.S. Brown.

AN EL-LABELING OF THE SUBGROUP LATTICE

In a 2001 paper, Shareshian conjectured that the subgroup lattice of a finite, solvable group has an EL-labeling. We construct such a labeling and verify that our labeling has the expected properties.

Shelling the coset poset

It is shown that the coset lattice of a finite group has shellable order complex if and only if the group is complemented. Furthermore, the coset lattice is shown to have a Cohen-Macaulay order complex in exactly the same conditions. The group theoretical tools used are relatively elementary, and avoid the classification of finite simple groups and of minimal finite simple groups.

Antichain Cutsets of Strongly Connected Posets

Rival and Zaguia showed that the antichain cutsets of a finite Boolean lattice are exactly the level sets. We show that a similar characterization of antichain cutsets holds for any strongly connected poset of locally finite height. As a corollary, we characterize the antichain cutsets in semimodular lattices, supersolvable lattices, Bruhat orders, locally shellable lattices, and many more. We also consider a generalization to strongly connected d-uniform hypergraphs.