Shellable graphs and sequentially Cohen-Macaulay bipartite graphs

Sequentially Cohen-Macaulay Graphs of Form θ_{n1, ...nk}

A characterization of shellable and sequentially Cohen-Macaulay

We consider a class of hypergraphs called hypercycles and we show that a hypercycle $C_n^{d,alpha}$ is shellable or sequentially the Cohen--Macaulay if and only if $nin{3,5}$. Also, we characterize Cohen--Macaulay hypercycles. These results are hypergraph versions of results proved for cycles in graphs.

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.

Some Algebraic and Combinatorial Properties of the Complete $T$-Partite Graphs

In this paper, we characterize the shellable complete $t$-partite graphs. We also show for these types of graphs the concepts vertex decomposable, shellable and sequentially Cohen-Macaulay are equivalent. Furthermore, we give a combinatorial condition for the Cohen-Macaulay complete $t$-partite graphs.

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.