Sequentially Cohen–Macaulay bipartite graphs: vertex decomposability and regularity

Sequentially Cohen-macaulay Bipartite Graphs: Vertex Decomposability and Regularity

Let G be a bipartite graph with edge ideal I(G) whose quotient ring R/I(G) is sequentially Cohen-Macaulay. We prove: (1) the independence complex of G must be vertex decomposable, and (2) the Castelnuovo-Mumford regularity of R/I(G) can be determined from the invariants of G.

Complement of Special Chordal Graphs and Vertex Decomposability

In this paper, we introduce a subclass of chordal graphs which contains $d$-trees and show that their complement are vertex decomposable and so is shellable and sequentially Cohen-Macaulay.

complement of special chordal graphs and vertex decomposability

in this paper, we introduce a subclass of chordal graphs which contains $d$-trees and show that their complement are vertex decomposable and so is shellable and sequentially cohen-macaulay.

Shellable graphs and sequentially Cohen-Macaulay bipartite graphs

Associated to a simple undirected graph G is a simplicial complex ∆G whose faces correspond to the independent sets of G. We call a graph G shellable if ∆G is a shellable simplicial complex in the non-pure sense of Björner-Wachs. We are then interested in determining what families of graphs have the property that G is shellable. We show that all chordal graphs are shellable. Furthermore, we cla...

Bounds for the Regularity of Edge Ideal of Vertex Decomposable and Shellable Graphs