Blowup algebras of ideals of vertex covers of bipartite graphs

Symbolic Powers of Ideals

“…We also discuss the equality I (n) = I n for square-free monomial ideals. This problem was related to a conjecture of Conforti and Cornuélos [CC90] on the max-flow and min-cut properties by Gitler, Valencia and Villarreal [GVV05] and Gitler, Reyes, and Villarreal [GRV05]. The Conforti-Cornuélos Conjecture is known in the context of symbolic powers as the Packing Problem (see Subsection 4.2), and it is a central problem in this theory.…”

Generalized cover ideals and the persistence property

“…When t = 1, astab(J 1 (Γ)) = 1 since Γ is bipartite. So the result follows from [9]. When t ≥ 2, let x be a vertex with deg x = ∆(Γ), i.e., a vertex of maximal degree.…”

Depth and Stanley depth of symbolic powers of cover ideals of graphs

“…Similarly sdepth S i+1 (S i+1 /(J(G) (k) ∩ S i+1 )) ≥ n − ν o (G) − 1. Now the claim follows by inequalities (3), (4), (5), and (6). Now, J ′ n = (J(G) (k) : x 1 x 2 .…”