Let G be a simple undirected graph and let Delta(G) be a simplicial complex whose faces correspond to the independent sets of G. A graph G is called shellable if Delta(G) is a shellable simplicial complex. We prove that the complement of a d-tree is a pure shellable graph. This generalizes a recent result of Ferrarello who used a theorem due to R. Froberg to prove that the complement of a d-tree is a Cohen-Macaulay graph. 

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

Introduction: Let G be a simple undirected graph over the vertex set V. Let I(G) denotes the edge ideal of G and DG be the simplicial complex whose faces correspond to the independent sets of G. This simplicial complex reflects many nice properties of G. A simplicial complex D is called shellable if the facets can given a linear order F1,…,Ft such that for all 1£t<f£s, there exists some vÎF1\Ft and some LÎ{1,…,f-1} with F1\Ft={v}. A result due to Hochster says that every pure shellable complex is Cohen-Macaulay over every field. A graph is called shellable, if the simplicial complex DG is a shellable simplicial complex.Aim: In this paper we focus on the question of what graphs G have the property that `G is Cohen-Macaulay, i.e. R/I(`G) is Cohen-Macaulay. We prove that the complement of a connected triangle-free graph is pure shellable and consequently Cohen-Macaulay.Methods: By providing an explicit shelling for the facets of D`G, whenever G is a connected triangle-free graph, we show `G, the complement of G, is pure shellable and consequently Cohen-Macaulay.Conclusion: The complement of any connected bipartite graph and any cycle is pure shellable and hence Cohen-Macaulay.

We give upper bounds for the regularity of edge ideal of some classes of graphs in terms of invariants of graph. We introduce two numbers a’ (G) and n (G) depending on graph G and show that for a vertex decomposable graphG, reg (R/I (G)) £ min{a’ (G), n (G)} and for a shellable graph G, reg (R/I (G))£n (G). Moreover, it is shown that for a graphG, where Gc is a d-tree, we have pd (R/I (G)) =maxvÎV (G) {degG (v)}.

In this paper, we characterize the shellable complete t-partite graphs. It is also shown that 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.

Let k be an integer greater than 2 and n1, . . . , nk be a sequence of positive integers with at most one of them being equal to 1. Let qn1,...,nk be a graph consisting of k paths, having only their endpoints in common. We characterize all sequentially Cohen-Macaulay graphs of this type. We also show for these types of graphs the notions of vertex decomposable, shellable and sequentially Cohen-Macaulay are equivalent.

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. This result improves the main result of Ferrarello who used a the-orem due to Froberg and extended a recent result of Dochtermann and Engstrom.