Regularity and Koszul property of symbolic powers of monomial ideals

Componentwise Linearity of Powers of Cover Ideals

“…As a consequence of our investigation, we partially answers the above question (Corollary 3.8, Corollary 4.4, Corollary 4.6). As an immediate consequence we recover and extend the main results of [9,17,26,30]. Dung, Hien, Nguyen, and Trung asked a following question: classify all star graphs based on a complete graph G such that all the symbolic powers of J(G) are componentwise linear, [9,Question 5.13].…”

“…Ordinary and symbolic powers of edge/cover ideals of graphs were studied by many authors (cf. [2,9,11,25,26,27,28,30,31]).…”

“…The study of edge ideals and cover ideals from both the algebraic and combinatorial points of view has become a very active area of research in commutative algebra (cf. [1,2,5,9,10,12,14,15,18,19,20,21,22,23,24,25,27,28,35]). A graph G is said to be vertex decomposable(shellable) if its independence complex ∆(G) has this property (see Section 2 for the definition).…”

“…Also, he characterized all graphs G with the property that J(G) (k) has a linear resolution for some (equivalently, for all) k ≥ 2, [32]. In [9], Dung, Hien, Nguyen and Trung proved that, all symbolic powers of the cover ideal of G ∪ W (V (G)), obtained from G by adding a whisker to each vertex in V (G), are componentwise linear. Then the first author of this paper proved that one only needs to whisker at the vertices of a vertex cover of G to get a new graph for which all symbolic powers of the cover ideal are componentwise linear, [30].…”

On the minimal free resolution of symbolic powers of cover ideals of graphs

“…As a consequence, we see that for any such graph, deg(J(G) (k) ) is a a linear function of k. Note that in general, deg(J(G) (k) ) is not a linear function (even eventually), as it is shown by Dung at al. [3,Theorem 5.15].…”