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PreprintCC BY 4.0arXiv

Homology of non-matching complexes under edge additions and applications to their Stanley-Reisner ideals

  • Jiawen Shan
  • Zexin Wang
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Abstract

For a bipartite graph G and an integer t≥2, let NMt(G) be its t-non-matching complex. We prove that adding an edge while preserving bipartiteness induces an injection on reduced homology in degree 2t−3. Combined with the cyclic-polytope model for non-matching complexes of cycles, this shows that NMt(G) has Leray number 2t−2 whenever G contains a cycle of length at least 2t. Under the same hypothesis, Hochster's formula yields regularity 2t−1 for the Stanley-Reisner ideal INMt(G), together with explicit lower bounds for the Betti numbers on its top regularity strand and for its projective dimension. If G contains a 2t-cycle, we also determine the maximal shifts of INMt(G) through homological degree |E(G)|−2t+1. For G=Kr,s with 2≤ t≤ r≤ s, we determine the depth, projective dimension, all maximal shifts, and the unique extremal Betti number of INMt(Kr,s), thereby settling a conjecture on facet ideals of chessboard complexes.

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Cite

BibTeXRIS
BibTeX
@misc{shan2026homology,
  title = {{Homology of non-matching complexes under edge additions and applications to their Stanley-Reisner ideals}},
  author = {Jiawen Shan and Zexin Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.02621},
  eprint = {2609.02621},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.02621},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Homology of non-matching complexes under edge additions and applications to their Stanley-Reisner ideals
AU  - Jiawen Shan
AU  - Zexin Wang
PY  - 2026
DA  - 2026/09/02
PB  - arXiv
DO  - 10.48550/arxiv.2609.02621
UR  - https://arxiv.org/abs/2609.02621
AB  - For a bipartite graph G and an integer t≥2, let NM_t(G) be its t-non-matching complex. We prove that adding an edge while preserving bipartiteness induces an injection on reduced homology in degree 2t−3. Combined with the cyclic-polytope model for non-matching complexes of cycles, this shows that NM_t(G) has Leray number 2t−2 whenever G contains a cycle of length at least 2t. Under the same hypothesis, Hochster's formula yields regularity 2t−1 for the Stanley-Reisner ideal I_NM_t(G), together with explicit lower bounds for the Betti numbers on its top regularity strand and for its projective dimension. If G contains a 2t-cycle, we also determine the maximal shifts of I_NM_t(G) through homological degree |E(G)|−2t+1. For G=K_r,s with 2≤ t≤ r≤ s, we determine the depth, projective dimension, all maximal shifts, and the unique extremal Betti number of I_NM_t(K_r,s), thereby settling a conjecture on facet ideals of chessboard complexes.
ER  -