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PreprintCC BY-NC-SA 4.0arXiv

Structural properties of admissible subgraphs and the depth of symbolic powers of cover ideals

  • Nguyen Thu Hang
  • Nguyen Thi Thanh Tam
  • Thanh Vu
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Abstract

Let G be a simple graph with cover ideal J(G) in a polynomial ring S = k[x1, …, xn]. For each integer t ≥ 1, let αt(G) denote the maximum size of an ordered matching M of G with alternating path length ℓ(M) ≤ 2t − 1. Hang, Tam, and Vu established that depth (S/J(G)(t)) ≤ n − 1 − αt(G) for all t ≥ 1, where J(G)(t) is the t-th symbolic power of J(G). In this paper, we introduce a natural hereditary class of graphs 𝒫k for which this inequality holds with equality for every t ≥ 1. We prove that 𝒫k contains all distance-hereditary graphs, and we show that within the class of chordal graphs, 𝒫k coincides with the class of distance-hereditary (or gem-free) graphs.

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Cite

BibTeXRIS
BibTeX
@misc{hang2026structural,
  title = {{Structural properties of admissible subgraphs and the depth of symbolic powers of cover ideals}},
  author = {Nguyen Thu Hang and Nguyen Thi Thanh Tam and Thanh Vu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.09316},
  eprint = {2610.09316},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.09316},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - Structural properties of admissible subgraphs and the depth of symbolic powers of cover ideals
AU  - Nguyen Thu Hang
AU  - Nguyen Thi Thanh Tam
AU  - Thanh Vu
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.09316
UR  - https://arxiv.org/abs/2610.09316
AB  - Let G be a simple graph with cover ideal J(G) in a polynomial ring S = k[x_1, …, x_n]. For each integer t ≥ 1, let α_t(G) denote the maximum size of an ordered matching M of G with alternating path length ℓ(M) ≤ 2t − 1. Hang, Tam, and Vu established that depth (S/J(G)^(t)) ≤ n − 1 − α_t(G) for all t ≥ 1, where J(G)^(t) is the t-th symbolic power of J(G). In this paper, we introduce a natural hereditary class of graphs 𝒫_k for which this inequality holds with equality for every t ≥ 1. We prove that 𝒫_k contains all distance-hereditary graphs, and we show that within the class of chordal graphs, 𝒫_k coincides with the class of distance-hereditary (or gem-free) graphs.
ER  -