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PreprintCC BY 4.0arXiv

Associated primes and v-numbers of integral closures of powers of edge ideals

  • Trung Chau
  • Tai Huy Ha
  • Thanh Vu
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Abstract

Let I = I(G) be the edge ideal of a finite simple graph G. We investigate the relationship between the associated primes and v-numbers of powers It and their integral closures It̅. For the third power, we establish the associated prime containment Ass(S/I3̅) ⊆ Ass(S/I3), which yields v(I3) ≤ v(I3̅). In contrast, for t ≥ 4, we show that Ass(S/It̅) and Ass(S/It) are in general incomparable. Furthermore, for every fixed t ≥ 3 and r≥ 4, we construct families of graphs for which the discrepancies v(It̅) − v(It) and v(Ir)−v(Ir̅) are arbitrarily large.

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Cite

BibTeXRIS
BibTeX
@misc{chau2026associated,
  title = {{Associated primes and v-numbers of integral closures of powers of edge ideals}},
  author = {Trung Chau and Tai Huy Ha and Thanh Vu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.31901},
  eprint = {2610.31901},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.31901},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Associated primes and v-numbers of integral closures of powers of edge ideals
AU  - Trung Chau
AU  - Tai Huy Ha
AU  - Thanh Vu
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.31901
UR  - https://arxiv.org/abs/2610.31901
AB  - Let I = I(G) be the edge ideal of a finite simple graph G. We investigate the relationship between the associated primes and v-numbers of powers I^t and their integral closures I^t̅. For the third power, we establish the associated prime containment Ass(S/I^3̅) ⊆ Ass(S/I^3), which yields v(I^3) ≤ v(I^3̅). In contrast, for t ≥ 4, we show that Ass(S/I^t̅) and Ass(S/I^t) are in general incomparable. Furthermore, for every fixed t ≥ 3 and r≥ 4, we construct families of graphs for which the discrepancies v(I^t̅) − v(I^t) and v(I^r)−v(I^r̅) are arbitrarily large.
ER  -