Symbolic Powers of Cover Ideals of Graphs and Koszul Property

Abstract

We show that attaching a whisker (or a pendant) at the vertices of a cycle cover of a graph results in a new graph with the following property: all symbolic powers of its cover ideal are Koszul or, equivalently, componentwise linear. This extends previous work where the whiskers were added to all vertices or to the vertices of a vertex cover of the graph.

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].…”

“…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]. Thereafter, the second author along with Gu and Hà generalized this result, proving that the same conclusion holds true after adding whiskers at the vertices of a cycle cover of the graph, [17].…”

“…In [17],the authors raised the following question: find a necessary and sufficient condition on a subset S of the vertices in a graph G such that J(G∪W (S)) (k) is componentwise linear where G ∪ W (S) is the graph obtained from G by adding a whisker to each vertex in S for all k ≥ 1. As a consequence of our investigation, we partially answers the above question (Corollary 3.8, Corollary 4.4, Corollary 4.6).…”