Decreasing behavior of the depth functions of edge ideals

Let I be the edge ideal of a connected non-bipartite graph and R the base polynomial ring. Then, depth R / I ≥ 1 and depth R / I t = 0 for t ≫ 1 . This paper studies the problem when depth R / I t = 1 for some t ≥ 1 and whether the depth function is non-increasing thereafter. Furthermore, we are abl...

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Bibliographic Details
Published in:Journal of algebraic combinatorics 2024-01, Vol.59 (1), p.37-53
Main Authors: Hien, Ha Thi Thu, Lam, Ha Minh, Trung, Ngo Viet
Format: Article
Language:English
Subjects:
Combinatorial analysis
Combinatorics
Computer Science
Convex and Discrete Geometry
Graph theory
Group Theory and Generalizations
Lattices
Mathematical analysis
Mathematics
Mathematics and Statistics
Order
Ordered Algebraic Structures
Polynomials
Rings (mathematics)
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Summary:Let I be the edge ideal of a connected non-bipartite graph and R the base polynomial ring. Then, depth R / I ≥ 1 and depth R / I t = 0 for t ≫ 1 . This paper studies the problem when depth R / I t = 1 for some t ≥ 1 and whether the depth function is non-increasing thereafter. Furthermore, we are able to give a simple combinatorial criterion for depth R / I ( t ) = 1 for t ≫ 1 and show that the condition depth R / I ( t ) = 1 is persistent, where I ( t ) denotes the t -th symbolic powers of I .
ISSN:0925-9899
1572-9192
DOI:10.1007/s10801-023-01278-8