[Paper Review] Symbolic powers of vertex cover ideals

S. Selvaraja|arXiv (Cornell University)|Aug 28, 2019

Commutative Algebra and Its Applications26 references4 citations

TL;DR

This paper establishes that symbolic powers of vertex cover ideals of certain vertex decomposable graphs—particularly those obtained by adding whiskers to a subset of vertices—have linear quotients. The key contribution is a characterization of when these symbolic powers are componentwise linear, leading to explicit formulas for Castelnuovo-Mumford regularity in terms of the maximum size of minimal vertex covers.

ABSTRACT

Let $G$ be a finite simple graph and $J(G)$ denote its cover ideal in a polynomial ring over a field $\mathbb{K}$. In this paper, we show that all symbolic powers of cover ideals of certain vertex decomposable graphs have linear quotients. Using these results, we give various conditions on a subset $S$ of the vertices of $G$ so that all symbolic powers of vertex cover ideals of $G \cup W(S)$, obtained from $G$ by adding a whisker to each vertex in $S$, have linear quotients. For instance, if $S$ is a vertex cover of $G$, then all symbolic powers of $J(G \cup W(S))$ have linear quotients. Moreover, we compute the Castelnuovo-Mumford regularity of symbolic powers of certain cover ideals.

Motivation & Objective

• To determine conditions under which symbolic powers of vertex cover ideals have linear quotients.
• To investigate how graph modifications—specifically adding whiskers to subsets of vertices—affect the algebraic properties of cover ideals.
• To compute the Castelnuovo-Mumford regularity of symbolic powers of cover ideals for specific graph classes.
• To extend known results on componentwise linearity to symbolic powers, particularly in the context of vertex decomposable and whiskered graphs.

Proposed method

• Utilizes vertex decomposability of graphs to infer linear quotients in symbolic powers of cover ideals.
• Applies the concept of symbolic powers as intersections of primary components corresponding to minimal primes of the ideal.
• Employs the Alexander duality between edge ideals and cover ideals to relate combinatorial graph properties to algebraic ideal properties.
• Uses the notion of componentwise linearity and linear quotients to analyze minimal free resolutions of symbolic powers.
• Applies results from combinatorial commutative algebra, including the characterization of regularity via highest-degree generators in componentwise linear ideals.
• Employs computational tools (Macaulay2 with EdgeIdeals, SimplicialDecomposability, SymbolicPowers packages) to verify and explore examples.

Experimental results

Research questions

• RQ1 Under what conditions do symbolic powers of cover ideals of whiskered graphs have linear quotients?
• RQ2 When does the symbolic power of a cover ideal remain componentwise linear after adding whiskers to a subset of vertices?
• RQ3 What is the Castelnuovo-Mumford regularity of symbolic powers of cover ideals for vertex decomposable or whiskered graphs?
• RQ4 For which vertex decomposable graphs is the k-th iterated whiskering construction also vertex decomposable for all k ≥ 1?
• RQ5 Does the regularity of symbolic powers of cover ideals equal k times the degree of the original cover ideal under certain graph conditions?

Key findings

• All symbolic powers of the cover ideal of $ G igcup W(S) $, where $ S $ is a vertex cover of $ G $, have linear quotients.
• If $ G $ is vertex decomposable and $ S eq ul $, then $ G igcup W(S) $ has symbolic powers of its cover ideal with linear quotients.
• For a star complete graph $ G $, $ ext{reg}(J(G)^{(k)}) = k imes ext{deg}(J(G)) $ for all $ k eq 1 $, with $ ext{deg}(J(G)) = |V(G)| - 1 $.
• For a graph $ G = H^ au $, where $ au $ is a clique vertex partition, $ ext{reg}(J(G)^{(k)}) = k imes ext{deg}(J(G)) $ for all $ k eq 1 $.
• For bipartite graphs satisfying the hypothesis of Theorem 4.3, $ J(G)^{(k)} = J(G)^k $, and $ ext{reg}(J(G)^{(k)}) = k imes ext{deg}(J(G)) $.
• In non-bipartite graphs, such as the one in Example 4.12, $ ext{reg}(J(G)^{(2)}) = 9 $, while $ ext{reg}(J(G)) = 4 $, showing that the regularity formula does not always extend.