2021/07/01 by S. Selvaraja, Selvaraja, S., Joseph W. Skelton +1
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2107.00739
openalex publication_date 2021/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Let G be a finite simple graph and J(G) denote its vertex cover ideal in a polynomial ring over a field. % \mathbbK. The k-th symbolic power of J(G) is denoted by J(G)(k). In this paper, we give a criteria for cover ideals of vertex decomposable graphs to have the property that all their symbolic powers are not componentwise linear. Also, we give a necessary and sufficient condition on G so that J(G)(k) is a componentwise linear ideal for some (equivalently, for all) k ≥ 2 when G is a graph such that G ∖ NG[A] has a simplicial vertex for any independent set A of G. Using this result, we prove that J(G)(k) is a componentwise linear ideal for several classes of graphs for all k ≥ 2. In particular, if G is a bipartite graph, then J(G) is a componentwise linear ideal if and only if J(G)k is a componentwise linear ideal for some (equivalently, for all) k ≥ 2.