72 research outputs found
Erdos-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 graph's 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.Comment: 14 pages; v2 has minor changes; v3 has further minor changes for
publicatio
Simplicial Trees are Sequentially Cohen-Macaulay
This paper uses dualities between facet ideal theory and Stanley-Reisner
theory to show that the facet ideal of a simplicial tree is sequentially
Cohen-Macaulay. The proof involves showing that the Alexander dual (or the
cover dual, as we call it here) of a simplicial tree is a componentwise linear
ideal. We conclude with additional combinatorial properties of simplicial
trees.Comment: 15 pages, 15 figure
- …