Cohen–Macaulay binomial edge ideals and accessible graphs

The cut sets of a graph are special sets of vertices whose removal disconnects the graph. They are fundamental in the study of binomial edge ideals, since they encode their minimal primary decomposition. We introduce the class of accessible graphs as the graphs with unmixed binomial edge ideal and whose cut sets form an accessible set system. We prove that the graphs whose binomial edge ideal is Cohen–Macaulay are accessible and we conjecture that the converse holds. We settle the conjecture for large classes of graphs, including chordal and traceable graphs, providing a purely combinatorial description of Cohen–Macaulayness. The key idea in the proof is to show that both properties are equivalent to a further combinatorial condition, which we call strong unmixedness.

Induced Matchings and the v-Number of Graded Ideals

We give a formula for the v-number of a graded ideal that can be used to compute this number. Then, we show that for the edge ideal I(G) of a graph G, the induced matching number of G is an upper bound for the v-number of I(G) when G is very well-covered, or G has a simplicial partition, or G is well-covered connected and contains neither four, nor five cycles. In all these cases, the v-number of I(G) is a lower bound for the regularity of the edge ring of G. We classify when the induced matching number of G is an upper bound for the v-number of I(G) when G is a cycle and classify when all vertices of a graph are shedding vertices to gain insight into the family of W2-graphs.

Upper bounds for the regularity of symbolic powers of certain classes of edge ideals

Let [Formula: see text] be a finite simple graph and [Formula: see text] denote the corresponding edge ideal in a polynomial ring over a field [Formula: see text]. In this paper, we obtain upper bounds for the Castelnuovo–Mumford regularity of symbolic powers of certain classes of edge ideals. We also prove that for several classes of graphs, the regularity of symbolic powers of their edge ideals coincides with that of their ordinary powers.

Unmixed and Cohen–Macaulay Weighted Oriented Kőnig Graphs

Let D be a weighted oriented graph, whose underlying graph is G, and let I (D) be its edge ideal. If G has no 3-, 5-, or 7-cycles, or G is Kőnig, we characterize when I (D) is unmixed. If G has no 3- or 5-cycles, or G is Kőnig, we characterize when I (D) is Cohen–Macaulay. We prove that I (D) is unmixed if and only if I (D) is Cohen–Macaulay when G has girth greater than 7 or G is Kőnig and has no 4-cycles.

Maximal independent sets and regularity of graphs

In this paper, we give a lower bound of the number of maximal independent sets in a graph [Formula: see text] in terms of the Castelnuovo–Mumford regularity of its edge ideal. We also find two classes of graphs achieving this minimum value.

Squarefree Powers of Edge Ideals of Forests

Let $I(G)^{[k]}$ denote the $k$th squarefree power of the edge ideal of $G$. When $G$ is a forest, we provide a sharp upper bound for the regularity of $I(G)^{[k]}$ in terms of the $k$-admissable matching number of $G$. For any positive integer $k$, we classify all forests $G$ such that $I(G)^{[k]}$ has linear resolution. We also give a combinatorial formula for the regularity of $I(G)^{[2]}$ for any forest $G$.

Betti Numbers of Weighted Oriented Graphs

Let $\mathcal{D}$ be a weighted oriented graph and $I(\mathcal{D})$ be its edge ideal. In this paper, we investigate the Betti numbers of $I(\mathcal{D})$ via upper-Koszul simplicial complexes, Betti splittings and the mapping cone construction. In particular, we provide recursive formulas for the Betti numbers of edge ideals of several classes of weighted oriented graphs. We also identify classes of weighted oriented graphs whose edge ideals have a unique extremal Betti number which allows us to compute the regularity and projective dimension for the identified classes. Furthermore, we characterize the structure of a weighted oriented graph $\mathcal{D}$ on $n$ vertices such that $\textrm{pdim } (R/I(\mathcal{D}))=n$ where $R=k[x_1,\ldots, x_n]$.

Edge Ideals with Regularity No More than Three

We prove that if G is a gap-free and chair-free simple graph, then the regularity of the edge ideal of G is no more than 3. If G is a gap-free and P4-free graph, then it is a chair-free graph; furthermore, the complement of G is chordal, and thus the regularity of G is 2.

ON THE DEPTH OF SYMBOLIC POWERS OF EDGE IDEALS OF GRAPHS

Assume that G is a graph with edge ideal $I(G)$ and star packing number $\alpha _2(G)$ . We denote the sth symbolic power of $I(G)$ by $I(G)^{(s)}$ . It is shown that the inequality $ \operatorname {\mathrm {depth}} S/(I(G)^{(s)})\geq \alpha _2(G)-s+1$ is true for every chordal graph G and every integer $s\geq 1$ . Moreover, it is proved that for any graph G, we have $ \operatorname {\mathrm {depth}} S/(I(G)^{(2)})\geq \alpha _2(G)-1$ .

Matching Numbers and the Regularity of the Rees Algebra of an Edge Ideal

The regularity $${\text {reg}}R(I(G))$$ reg R ( I ( G ) ) of the Rees ring R(I(G)) of the edge ideal I(G) of a finite simple graph G is studied. It is shown that, if R(I(G)) is normal, one has $${\text {mat}}(G) \le {\text {reg}}R(I(G)) \le {\text {mat}}(G) + 1$$ mat ( G ) ≤ reg R ( I ( G ) ) ≤ mat ( G ) + 1 , where $${\text {mat}}(G)$$ mat ( G ) is the matching number of G. In general, the induced matching number is a lower bound for the regularity, which can be shown by applying the squarefree divisor complex.