Cohen-Macaulay Graphs Research Articles

INDEPENDENT SETS FROM AN ALGEBRAIC PERSPECTIVE

In this paper, we study the basic problem of counting independent sets in a graph and, in particular, the problem of counting antichains in a finite poset, from an algebraic perspective. We show that neither independence polynomials of bipartite Cohen–Macaulay graphs nor Hilbert series of initial ideals of radical zero-dimensional complete intersections ideals, can be evaluated in polynomial time, unless #P = P. Moreover, we present a family of radical zero-dimensional complete intersection ideals JP associated to a finite poset P, for which we describe a universal Gröbner basis. This implies that the bottleneck in computing the dimension of the quotient by JP (that is, the number of zeros of JP) using Gröbner methods lies in the description of the standard monomials.

Cohen–Macaulay Graphs and Face Vectors of Flag Complexes

We introduce a construction on a flag complex that by means of modifying the associated graph generates a new flag complex whose h-vector is the face vector of the original complex. This construction yields a vertex-decomposable, hence Cohen–Macaulay, complex. From this we get a (nonnumerical) characterization of the face vectors of flag complexes and deduce also that the face vector of a flag complex is the h-vector of some vertex-decomposable flag complex. We conjecture that the converse of the latter is true and prove this, by means of an explicit construction, for h-vectors of Cohen–Macaulay flag complexes arising from bipartite graphs. We also give several new characterizations of bipartite graphs with Cohen–Macaulay or Buchsbaum independence complexes.

Sequentially $S_{r}$ simplicial complexes and sequentially $S_{2}$ graphs

We introduce sequentially S r modules over a commutative graded ring and sequentially S r simplicial complexes. This generalizes two properties for modules and simplicial complexes: being sequentially Cohen-Macaulay, and satisfying Serre's condition S r . In analogy with the sequentially Cohen-Macaulay property, we show that a simplicial complex is sequentially S r if and only if its pure i-skeleton is S r for all i. For r = 2, we provide a more relaxed characterization. As an algebraic criterion, we prove that a simplicial complex is sequentially S r if and only if the minimal free resolution of the ideal of its Alexander dual is componentwise linear in the first r steps. We apply these results for a graph, i.e., for the simplicial complex of the independent sets of vertices of a graph. We characterize sequentially S r cycles showing that the only sequentially S 2 cycles are odd cycles and, for r > 3, no cycle is sequentially S r with the exception of cycles of length 3 and 5. We extend certain known results on sequentially Cohen-Macaulay graphs to the case of sequentially S r graphs. We prove that a bipartite graph is vertex decomposable if and only if it is sequentially S 2 . We provide some more results on certain graphs which in particular implies that any graph with no chordless even cycle is sequentially S 2 . Finally, we propose some questions.

An alternative proof of a characterization of Cohen-Macaulay bipartite graphs

Recently, Herzog and Hibi explicitly described all Cohen-Macaulay bipartite graphs by using the Stanley-Reisner ideal of the Alexander dual of the simplicial complex Δ P associated to a finite poset P. In this paper, we will present a short proof that does not use the Stanley-Reisner ideal of the Alexander dual of Δ P .

Whiskers and sequentially Cohen–Macaulay graphs

We investigate how to modify a simple graph G combinatorially to obtain a sequentially Cohen–Macaulay graph. We focus on adding configurations of whiskers to G, where to add a whisker one adds a new vertex and an edge connecting this vertex to an existing vertex of G. We give various sufficient conditions and necessary conditions on a subset S of the vertices of G so that the graph G ∪ W ( S ) , obtained from G by adding a whisker to each vertex in S, is a sequentially Cohen–Macaulay graph. For instance, we show that if S is a vertex cover of G, then G ∪ W ( S ) is a sequentially Cohen–Macaulay graph. On the other hand, we show that if G ∖ S is not sequentially Cohen–Macaulay, then G ∪ W ( S ) is not a sequentially Cohen–Macaulay graph. Our work is inspired by and generalizes a result of Villarreal on the use of whiskers to get Cohen–Macaulay graphs.

Sequentially Cohen-Macaulay edge ideals

Let G be a simple undirected graph on n vertices, and let I(G) C R = k[x 1 ,...,x n ] denote its associated edge ideal. We show that all chordal graphs G are sequentially Cohen-Macaulay; our proof depends upon showing that the Alexander dual of I(G) is componentwise linear. Our result complements Faridi's theorem that the facet ideal of a simplicial tree is sequentially Cohen-Macaulay and implies Herzog, Hibi, and Zheng's theorem that a chordal graph is Cohen-Macaulay if and only if its edge ideal is unmixed. We also characterize the sequentially Cohen-Macaulay cycles and produce some examples of nonchordal sequentially Cohen-Macaulay graphs.

The complement of a d-tree is Cohen-Macaulay

In this work we obtain the result that the complement of a $d$-tree is a Cohen-Macaulay graph. To do this we use a theorem by Fröberg that estabilishes a condition for a Stanley-Reisner ring of a simplicial complex to be Cohen-Macaulay and a useful lemma to pass from a Stanley-Reisner ideal of a simplicial complex to an edge ideal of a graph.

Cohen-macaulay graphs

For a graph G we consider its associated ideal I(G). We uncover large classes of Cohen-Macaulay (=CM) graphs, in particular the full subclass of CM trees is presented. A formula for the Krull dimension of the symmetric algebra of I(G) is given along with a description of when this algebra is a domain. The first Koszul homology module of a CM tree is also studied.