25 research outputs found
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.Comment: 13 pages, 3 figures. v2: Improved exposition, added Section 5.2 and
additional references. v3: minor corrections for publicatio
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.Comment: 12 pages; v4 has minor changes for publicatio
A new subgroup lattice characterization of finite solvable groups
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 that of 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.Comment: 15 pages; v2 has minor changes for publication; v3 minor typos fixe
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.Comment: 8 pages. v2 has minor exposition improvements and typos fixed, and
one reference added. v3 has typos fixe