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Edge ideals of complements of a class of circular interval graphs

  • Bilal Ahmad Wani
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Abstract

Let Cnm be the m-th power of the cycle on n vertices, and let Gnm be its complement, where m≥1 and n≥3m+1. We prove that every induced subcomplex of the clique complex of Cnm is homotopy equivalent either to a disjoint union of contractible complexes or to a circle, and we establish this dichotomy for a larger class of circular interval graphs. As a consequence, we obtain a closed formula for the linear strand of the Betti table of R/I(Gnm). We also prove that R/I(Gnm) is a Buchsbaum ring of depth $2$ and that the independence complex of Gnm is a triangulated m-manifold with boundary for m≥2, namely a Möbius band or an annulus when m=2. For the complement H̅ of every graph H in the above class, we show that reg(I(H̅)k)=2k+indm(H̅)−1 for all k≥2, where indm denotes the induced matching number. In particular, some power of I(Gnm) has a linear resolution if and only if n≥4m+1. In this range, we further prove that the colon ideals (Ik+1:M), where I=I(Gnm) and M is a minimal monomial generator of Ik, have regularity $2$.

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Cite

BibTeXRIS
BibTeX
@misc{wani2026edge,
  title = {{Edge ideals of complements of a class of circular interval graphs}},
  author = {Bilal Ahmad Wani},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.31105},
  eprint = {2609.31105},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.31105},
}
RIS
TY  - UNPB
TI  - Edge ideals of complements of a class of circular interval graphs
AU  - Bilal Ahmad Wani
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.31105
UR  - https://arxiv.org/abs/2609.31105
AB  - Let C_n^m be the m-th power of the cycle on n vertices, and let Gnm be its complement, where m≥1 and n≥3m+1. We prove that every induced subcomplex of the clique complex of C_n^m is homotopy equivalent either to a disjoint union of contractible complexes or to a circle, and we establish this dichotomy for a larger class of circular interval graphs. As a consequence, we obtain a closed formula for the linear strand of the Betti table of R/I(Gnm). We also prove that R/I(Gnm) is a Buchsbaum ring of depth $2$ and that the independence complex of Gnm is a triangulated m-manifold with boundary for m≥2, namely a Möbius band or an annulus when m=2. For the complement H̅ of every graph H in the above class, we show that reg(I(H̅)^k)=2k+indm(H̅)−1 for all k≥2, where indm denotes the induced matching number. In particular, some power of I(Gnm) has a linear resolution if and only if n≥4m+1. In this range, we further prove that the colon ideals (I^k+1:M), where I=I(Gnm) and M is a minimal monomial generator of I^k, have regularity $2$.
ER  -