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Weak and strong Lefschetz properties for vertex cover Artinian algebras associated to graphs

  • Trung Chau
  • Tran Quang Hoa
  • Dang Thi Kieu Uyen
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Abstract

Let G be a finite simple graph and let Ac(G) be the Artinian algebra associated with its cover ideal. We prove that Ac(G) has the WLP when τ(G)>|V(G)|/2, where τ(G) denotes the size of a minimum vertex cover of G. As a consequence, Ac(G) has the WLP with high probability when the Erdős-Rényi random graph model is considered. Moreover, we study the borderline case τ(G)=|V(G)|/2 and as a result, classify the WLP for paths, cycles, Ferrers graphs, and well-covered trees.

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Cite

BibTeXRIS
BibTeX
@misc{chau2026weak,
  title = {{Weak and strong Lefschetz properties for vertex cover Artinian algebras associated to graphs}},
  author = {Trung Chau and Tran Quang Hoa and Dang Thi Kieu Uyen},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.36674},
  eprint = {2609.36674},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.36674},
}
RIS
TY  - UNPB
TI  - Weak and strong Lefschetz properties for vertex cover Artinian algebras associated to graphs
AU  - Trung Chau
AU  - Tran Quang Hoa
AU  - Dang Thi Kieu Uyen
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.36674
UR  - https://arxiv.org/abs/2609.36674
AB  - Let G be a finite simple graph and let A_c(G) be the Artinian algebra associated with its cover ideal. We prove that A_c(G) has the WLP when τ(G)>|V(G)|/2, where τ(G) denotes the size of a minimum vertex cover of G. As a consequence, A_c(G) has the WLP with high probability when the Erdős-Rényi random graph model is considered. Moreover, we study the borderline case τ(G)=|V(G)|/2 and as a result, classify the WLP for paths, cycles, Ferrers graphs, and well-covered trees.
ER  -