Comparing powers of edge ideals

Abstract

Given a nontrivial homogeneous ideal I ⊆ k[x 1 , x 2 , . . . , x d ], a problem of great recent interest has been the comparison of the rth ordinary power of I and the mth symbolic power I (m) . This comparison has been undertaken directly via an exploration of which exponents m and r guarantee the subset containment I (m) ⊆ I r and asymptotically via a computation of the resurgence ρ(I), a number for which any m/r > ρ(I) guarantees I (m) ⊆ I r . Recently, a third quantity, the symbolic defect, was introduced; as I t ⊆ I (t) , the symbolic defect is the minimal number of generators required to add to I t in order to get I (t) .We consider these various means of comparison when I is the edge ideal of certain graphs by describing an ideal J for which I (t) = I t + J. When I is the edge ideal of an odd cycle, our description of the structure of I (t) yields solutions to both the direct and asymptotic containment questions, as well as a partial computation of the sequence of symbolic defects.

Invariants of the symbolic powers of edge ideals

“…In the next lemma, the elements in the symbolic power of edge ideals have been described in terms of minimal vertex cover of the graph. Using Lemma 2.4 and the concept of vertex weight Janssen et al in [9] described the elements of symbolic powers of edge ideals as follows…”

“…In this case we will try to understand the generators of (L(t)). For this we recall the definition of the optimal form introduced in [9].…”

“…In [14] Simis, Vasconcelos and Villarreal have proved that G is a bipartite graph if and only if I s = I (s) , for every s ∈ N. So it is interesting to study the symbolic powers of edge ideal of non-bipartite graphs. The class of non-bipartite graph which was studied first is odd cycle by Janssen et al in [9]. They have described the generators of the symbolic powers of the edge ideal of an odd cycle by using the concept of minimal vertex cover and calculated the invariants associated to the symbolic powers of the edge ideal of the same graph.…”

“…The work of this paper is mainly motivated by the papers [9] and [5]. In this paper we study the structure of the symbolic powers of edge ideal of clique sum of two different odd length cycles joined at a single vertex and prove Minh's conjecture for the class of complete graphs.…”

“…In section 2, we recall all the definitions and results that will be required for the rest of the paper. Motivated by the concept of vertex weight described in [9] by Janssen, in section 3, we find the generators of the symbolic powers of edge ideal of clique sum of two different odd length cycles joined at a single vertex by explicitly choosing a minimal vertex cover. Using the description of the generating set we compute the Waldschmidt constant.…”