Cohen-Macaulay Graphs Research Articles

On the Castelnuovo–Mumford regularity of squarefree powers of edge ideals

Assume that G is a graph with edge ideal I(G) and matching number match(G). For every integer s≥1, we denote the s-th squarefree power of I(G) by I(G)[s]. It is shown that for every positive integer s≤match(G), the inequality reg(I(G)[s])≤match(G)+s holds provided that G belongs to either of the following classes: (i) very well-covered graphs, (ii) semi-Hamiltonian graphs, or (iii) sequentially Cohen-Macaulay graphs. Moreover, we prove that for every Cameron-Walker graph G and for every positive integer s≤match(G), we have reg(I(G)[s])=match(G)+s.

Correction to: Powers of the vertex cover ideals (Collect. Math. 65 (2014) 169–181)

We describe a combinatorial condition on a graphwhich guarantees that all powers of its vertex cover ideal are componentwise linear. Then motivated by Eagon and Reiner’s Theorem we study whether all powers of the vertex cover ideal of a Cohen-Macaulay graph have linear free resolutions. After giving a complete characterization of Cohen-Macaulay cactus graphs (i.e., connected graphs in which each edge belongs to at most one cycle) we show that all powers of their vertex cover ideals have linear resolutions.

Some Cohen–Macaulay graphs arising from finite commutative rings

For a given finite commutative ring [Formula: see text] with [Formula: see text], one may associate a graph which is called the total graph of [Formula: see text]. This graph has [Formula: see text] as the vertex set and its two distinct vertices [Formula: see text] and [Formula: see text] are adjacent exactly whenever [Formula: see text] is a zero-divisor of [Formula: see text]. In this paper, we give necessary and sufficient conditions for two classes of total graphs to be Cohen–Macaulay.

A Construction of Sequentially Cohen–Macaulay Graphs

For every simple graph [Formula: see text], a class of multiple clique cluster-whiskered graphs [Formula: see text] is introduced, and it is shown that all such graphs are vertex decomposable; thus, the independence simplicial complex [Formula: see text] is sequentially Cohen–Macaulay. The properties of the graphs [Formula: see text] and [Formula: see text] constructed by Cook and Nagel are studied, including the enumeration of facets of the complex [Formula: see text] and the calculation of Betti numbers of the cover ideal [Formula: see text]. We also prove that the complex[Formula: see text] is strongly shellable and pure for either a Boolean graph [Formula: see text] or the full clique-whiskered graph [Formula: see text] of [Formula: see text], which is obtained by adding a whisker to each vertex of [Formula: see text]. This implies that both the facet ideal [Formula: see text] and the cover ideal [Formula: see text] have linear quotients.

Cohen–Macaulayness of a class of graphs versus the class of their complements

Let n≥2 be an integer. The graph G(n) is obtained by letting all the elements of {0,…,n−1} to be the vertices and defining distinct vertices x and y to be adjacent if and only if gcd⁡(x+y,n)=1. In this paper, well-coveredness, Cohen–Macaulayness, vertex-decomposability and Gorensteinness of these graphs and their complements are characterized. These characterizations provide large classes of Cohen–Macaulay and non Cohen–Macaulay graphs.

A Construction of Cohen-Macaulay Graphs

Abstract We present a technique to construct Cohen–Macaulay graphs from a given graph; if this graph fulfills certain conditions. As a consequence, we characterize Cohen–Macaulay paths.

Prime graphs, matchings and the Castelnuovo-Mumford regularity

We demonstrate the effectiveness of prime graphs for the calculation of the (Castelnuovo-Mumford) regularity of graphs. Such a notion allows us to reformulate the regularity as a generalized induced matching problem and perform regularity calculations in specific graph classes, including $(C_3,P_5)$-free graphs, $P_6$-free bipartite graphs and all Cohen-Macaulay graphs of girth at least five. In particular, we verify that the five cycle graph $C_5$ is the unique connected graph satisfying the inequality $im (G)\lt \mbox {reg}(G)=m (G)$. In addition, we prove that, for each integer $n\geq 1$, there exists a vertex decomposable perfect prime graph $G_n$ with $\mbox {reg}(G_n)=n$.

Shedding vertices of vertex decomposable well-covered graphs

We focus our attention on well-covered graphs that are vertex decomposable. We show that for many known families of these vertex decomposable graphs, the set of shedding vertices forms a dominating set. We then construct three new infinite families of well-covered graphs, none of which have this property. We use these results to provide a minimal counterexample to a conjecture of Villarreal regarding Cohen–Macaulay graphs.

Improved bounds for the regularity of edge ideals of graphs

Let G be a graph with n vertices, let \(S={\mathbb {K}}[x_1,\dots ,x_n]\) be the polynomial ring in n variables over a field \({\mathbb {K}}\) and let I(G) denote the edge ideal of G. For every collection \({\mathcal {H}}\) of connected graphs with \(K_2\in {\mathcal {H}}\), we introduce the notions of \({{\mathrm{ind-match}}}_{{\mathcal {H}}}(G)\) and \({{\mathrm{min-match}}}_{{\mathcal {H}}}(G)\). It will be proved that the inequalities \({{\mathrm{ind-match}}}_{\{K_2, C_5\}}(G)\le \mathrm{reg}(S/I(G))\le {{\mathrm{min-match}}}_{\{K_2, C_5\}}(G)\) are true. Moreover, we show that if G is a Cohen–Macaulay graph with girth at least five, then \(\mathrm{reg}(S/I(G))={{\mathrm{ind-match}}}_{\{K_2, C_5\}}(G)\). Furthermore, we prove that if G is a paw-free and doubly Cohen–Macaulay graph, then \(\mathrm{reg}(S/I(G))={{\mathrm{ind-match}}}_{\{K_2, C_5\}}(G)\) if and only if every connected component of G is either a complete graph or a 5-cycle graph. Among other results, we show that for every doubly Cohen–Macaulay simplicial complex, the equality \(\mathrm{reg}({\mathbb {K}}[\Delta ])=\mathrm{dim}({\mathbb {K}}[\Delta ])\) holds.

The arithmetical rank of the edge ideals of cactus graphs

ABSTRACTWe prove that, for the edge ideal of a cactus graph, the arithmetical rank is bounded above by the sum of the number of cycles and the maximum height of its associated primes. The bound is sharp, but in many cases, it can be improved. Moreover, we show that the edge ideal of a Cohen–Macaulay graph that contains exactly one cycle or is chordal or has no cycles of length 4 and 5 is a set-theoretic complete intersection.

Bounds on degrees of birational maps with arithmetically Cohen–Macaulay graphs

A rational map whose source and image are projectively embedded varieties is said to have an arithmetically Cohen–Macaulay graph if the Rees algebra of one (hence any) of its base ideals is a Cohen–Macaulay ring. The main objective of this paper is to obtain an upper bound for the degree of a representative of the map in case it is birational onto the image with an arithmetically Cohen–Macaulay graph. In the plane case a complete classification is given of the Cremona maps with arithmetically Cohen–Macaulay graph, while in arbitrary dimension n it is shown that a Cremona map with arithmetically Cohen–Macaulay graph has degree at most n2.

Cohen Macaulayness and Arithmetical Rank of Generalized Theta Graphs

In this paper, we study some algebraic invariants of the edge ideal of generalized theta graphs, such as arithmetical rank, big height and height. We give an upper bound for the difference between the arithmetical rank and big height. Moreover, all Cohen-Macaulay (and unmixed) graphs of this type will be characterized.

A Note on Cohen–Macaulay Graphs

We show that the edge ideal of a Cohen–Macaulay graph on 2n non-isolated vertices, whose height is n, is always a set-theoretic complete intersection. This result, in particular, applies to all Cohen–Macaulay bipartite graphs.

A New Construction for Cohen–Macaulay Graphs

Let G be a finite simple graph on a vertex set V(G) = {x 11,…, x n1}. Also let m 1,…, m n ≥ 2 be integers and G 1,…, G n be connected simple graphs on the vertex sets V(G i ) = {x i1,…, x im i }. In this article, we provide necessary and sufficient conditions on G 1,…, G n for which the graph obtained by attaching the G i to G is unmixed or vertex decomposable. Then we characterize Cohen–Macaulay and sequentially Cohen–Macaulay graphs obtained by attaching the cycle graphs or connected chordal graphs to arbitrary graphs.

Cohen–Macaulay graphs with large girth

We classify Cohen–Macaulay graphs of girth at least 5 and planar Gorenstein graphs of girth at least 4. Moreover, such graphs are also vertex decomposable.

Cohen-Macaulay bipartite graphs in arbitrary codimension

Let G G be an unmixed bipartite graph of dimension d − 1 d-1 . Assume that K n , n K_{n,n} , with n ≥ 2 n\ge 2 , is a maximal complete bipartite subgraph of G G of minimum dimension. Then G G is Cohen-Macaulay in codimension t t if and only if t ≥ d − n + 1 t\ge d-n+1 . This is derived from a characterization of Cohen-Macaulay bipartite graphs by Herzog and Hibi and generalizes a recent result of Cook and Nagel on unmixed Buchsbaum graphs. Furthermore, we show that any unmixed bipartite graph G G which is Cohen-Macaulay in codimension t t , is obtained from a Cohen-Macaulay graph by replacing certain edges of G G with complete bipartite graphs. Thus, in light of combinatorial characterization of Cohen-Macaulay bipartite graphs, our result may be considered purely combinatorial.

Very well-covered graphs and their h-vectors

Let G be a Cohen–Macaulay very well-covered graph. We prove that the h-vector of the independence complex of G is precisely the f-vector of a flag complex. Moreover, we show that the h-vector of clique-whiskered graphs is exactly the h-vector of Cohen–Macaulay very well-covered graphs. In particular, we show that Kalai’s conjecture holds if G is a very well covered Cohen–Macaulay graph.

Cohen–Macaulay Circulant Graphs

Let G be the circulant graph C n (S) with , and let I(G) denote the edge ideal in the ring R = k[x 1,…, x n ]. We consider the problem of determining when G is Cohen–Macaulay, i.e, R/I(G) is a Cohen–Macaulay ring. Because a Cohen–Macaulay graph G must be well-covered, we focus on known families of well-covered circulant graphs of the form C n (1, 2,…, d). We also characterize which cubic circulant graphs are Cohen–Macaulay. We end with the observation that even though the well-covered property is preserved under lexicographical products of graphs, this is not true of the Cohen–Macaulay property.

Vertex-Decomposable Graphs, Codismantlability, Cohen-Macaulayness, and Castelnuovo-Mumford Regularity

We call a vertex $x$ of a graph $G=(V,E)$ a codominated vertex if $N_G[y]\subseteq N_G[x]$ for some vertex $y\in V\backslash \{x\}$, and a graph $G$ is called codismantlable if either it is an edgeless graph or it contains a codominated vertex $x$ such that $G-x$ is codismantlable. We show that $(C_4,C_5)$-free vertex-decomposable graphs are codismantlable, and prove that if $G$ is a $(C_4,C_5,C_7)$-free well-covered graph, then vertex-decomposability, codismantlability and Cohen-Macaulayness for $G$ are all equivalent. These results complement and unify many of the earlier results on bipartite, chordal and very well-covered graphs. We also study the Castelnuovo-Mumford regularity $reg(G)$ of such graphs, and show that $reg(G)=im(G)$ whenever $G$ is a $(C_4,C_5)$-free vertex-decomposable graph, where $im(G)$ is the induced matching number of $G$. Furthermore, we prove that $H$ must be a codismantlable graph if $im(H)=reg(H)=m(H)$, where $m(H)$ is the matching number of $H$. We further describe an operation on digraphs that creates a vertex-decomposable and codismantlable graph from any acyclic digraph. By way of application, we provide an infinite family $H_n$ ($n\geq 4$) of sequentially Cohen-Macaulay graphs whose vertex cover numbers are half of their orders, while containing no vertex of degree-one such that they are vertex-decomposable, and $reg(H_n)=im(H_n)$ if $n\geq 6$. This answers a recent question of Mahmoudi et al.

Powers of the vertex cover ideals

We describe a combinatorial condition on a graph which guarantees that all powers of its vertex cover ideal are componentwise linear. Then motivated by Eagon and Reiner’s Theorem we study whether all powers of the vertex cover ideal of a Cohen-Macaulay graph have linear free resolutions. After giving a complete characterization of Cohen-Macaulay cactus graphs (i.e., connected graphs in which each edge belongs to at most one cycle) we show that all powers of their vertex cover ideals have linear resolutions.