Certain classes of Cohen–Macaulay multipartite graphs

Abstract

The Cohen-Macaulay property of a graph arising from a poset has been studied by various authors. In this article, we study the Cohen-Macaulay property of a graph arising from a family of reflexive and antisymmetric relations on a set. We use this result to find classes of multipartite graphs which are Cohen-Macaulay. IntroductionGraphs and simplicial complexes play an important role in combinatorial commutative algebra. In order to see the relationship between commutative algebra and combinatorics, one can associate monomial ideals to graphs or simplicial complexes. Many authors have studied the connection between the algebraic properties of these ideals and the combinatorial properties of the corresponding combinatorial objects, see [4, Chapter 9]. In this article, our main focus is to study the edge ideal of a graph. A graph is called Cohen-Macaulay if the corresponding edge ideal is Cohen-Macaulay.The Cohen-Macaulay property of graphs has been well studied for various classes. Herzog-Hibi ([3]) prove that a bipartite graph is Cohen-Macaulay if and only if it is arising from a poset. For a finite poset and r, s ∈ N, Ene-Herzog-Mohammadi ([2]) associated a monomial ideal generated in degree s, to the set of all multichains of length r in a poset, and proved that this ideal is Cohen-Macaulay. Note that if s = 2, then these ideals are edge ideals of some r-partite graphs. Motivated by these results, we associate a monomial ideal to a family of posets, and find a class of Cohen-Macaulay r-partite graphs.Our main tool in this article is the following well known relationship between the Stanley-Reisner ideal and its Alexander dual: The Stanley-Reisner ideal is Cohen-Macaulay if and only if its Alexander dual has a linear resolution. For more details see [1, Theorem 3].This paper has been organized in the following manner. In Section 2, we introduce the basic notions which are used throughout the article, more details can be found in [4]. In Section 3, we associate a monomial ideal H r (P) to a family P of partial order relations on a finite set. In Lemma 3.3, we prove that monomial ideal H r (P) has a linear resolution. This forces the Alexander dual of H r (P) to be Cohen-Macaulay.Section 4 is devoted to finding the classes of Cohen-Macaulay r-partite graphs. In Theorem 4.4, we see that the Alexander dual of H r (P) is an edge ideal of an r-partite graph associated to a family of reflexive and antisymmetric relations on a given set. Using this we find classes of Cohen-Macaulay graphs which are recorded in Theorems 4.7 and 4.10. Preliminaries2.1. Notation. The following notation is used throughout the article.

Symbolic powers of certain cover ideals of graphs

“…We now consider the complete multipartite graphs. The resolution of powers, Cohen-Macaulayness of cover ideals of complete multipartite graphs and vanishing ideal of the parametrized algebraic toric set associated to the complete multipartite graphs have already been looked into by several authors, [19], [22], [23], [24]. We prove that, if G is complete multipartite with partition…”