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PreprintCC BY 4.0arXiv

Projective dimension of closed neighborhood hypergraphs via extended double covers

  • Yusuf Civan
  • Anurag Singh
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Abstract

Let G be a finite and simple graph without isolated vertices. We investigate the projective dimension of the closed neighborhood hypergraph 𝒩[G] and its relationship with the Castelnuovo-Mumford regularity of the extended bipartite double cover Be(G) of G. We establish the general upper bound prod−dim (𝒩[G]) ≤ reg(Be(G)) for all graphs. Furthermore, we prove that the exact equalities prod−dim (𝒩[G]) = reg(Be(G)) =α(G) hold when G belongs to several prominent graph classes, including König-Egerváry (contains all bipartite graphs), cographs, co-chordal, chordal and comparability graphs, where α(G) denotes the independence number. Our method of proofs relies on connecting algebraic invariants to the underlying combinatorial structure of graphs through covering, domination and matching parameters, together with the use of homology tools.

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Cite

BibTeXRIS
BibTeX
@misc{civan2026projective,
  title = {{Projective dimension of closed neighborhood hypergraphs via extended double covers}},
  author = {Yusuf Civan and Anurag Singh},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.06556},
  eprint = {2610.06556},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.06556},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Projective dimension of closed neighborhood hypergraphs via extended double covers
AU  - Yusuf Civan
AU  - Anurag Singh
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.06556
UR  - https://arxiv.org/abs/2610.06556
AB  - Let G be a finite and simple graph without isolated vertices. We investigate the projective dimension of the closed neighborhood hypergraph 𝒩[G] and its relationship with the Castelnuovo-Mumford regularity of the extended bipartite double cover B_e(G) of G. We establish the general upper bound prod−dim (𝒩[G]) ≤ reg(B_e(G)) for all graphs. Furthermore, we prove that the exact equalities prod−dim (𝒩[G]) = reg(B_e(G)) =α(G) hold when G belongs to several prominent graph classes, including König-Egerváry (contains all bipartite graphs), cographs, co-chordal, chordal and comparability graphs, where α(G) denotes the independence number. Our method of proofs relies on connecting algebraic invariants to the underlying combinatorial structure of graphs through covering, domination and matching parameters, together with the use of homology tools.
ER  -