The v-numbers and linear presentations of ideals of covers of graphs
Humberto Mu\~noz-George, Enrique Reyes, Rafael H. Villarreal

TL;DR
This paper investigates the v-number and linear presentation properties of cover ideals of graphs, providing classifications and conditions based on graph combinatorics and algebraic properties, especially for unmixed and K"onig graphs.
Contribution
It offers a comprehensive classification of when cover ideals have linear presentations and determines v-numbers using combinatorial graph properties, extending previous algebraic results.
Findings
Classifies v-number extremal values via vertex cover exchange properties.
Expresses v-number in terms of covering number for linearly presented cover ideals.
Provides conditions for the connectivity of the graph of minimal vertex covers.
Abstract
Let be a graph and let be its ideal of covers. The aims of this work are to study the {\rm v}-number of and to study when is linearly presented using combinatorics and commutative algebra. We classify when attains its minimum and maximum possible values in terms of the vertex covers of the graph that satisfy the exchange property. If the cover ideal of a graph has a linear presentation, we express its v-number in terms of the covering number of the graph. If is unmixed, the graph of is the graph whose vertices are the minimal vertex covers of and whose edges are the pairs such that . We show necessary and sufficient conditions for the graph of to be connected. Then, for unmixed K\"onig graphs, we classify when is linearly presented using graph theory, and…
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Taxonomy
TopicsCommutative Algebra and Its Applications · Rings, Modules, and Algebras · Polynomial and algebraic computation
