PreprintCC BY 4.0arXiv
Homology of matching complexes of 3× n grid graphs
Abstract
For a finite simple graph G, the matching complex M(G) is the simplicial complex whose vertex set is the edge set of G and whose simplices are all the matchings in G. The topology of the matching complex of the m× n grid graph Gm× n is known only for m = 1,2, in which cases it is homotopy equivalent to a wedge of spheres. In this article, we study the matching complex M(G3 × n). We prove that for n≥2, its reduced homology vanishes in dimensions i ≤ n−2 and in top dimension, while H̃n−1(M(G3× n))≠ 0. We also show that M(G3 × n) is simply connected for n ≥ 3. Consequently, the topological connectivity of M(G3× n) is n−2.
Related
- Homology of non-matching complexes under edge additions and applications to their Stanley-Reisner ideals
- Integral homology lattices of matching and circulant complexes
- The rational homology of independence complexes of Kneser graphs on triples
- The Homotopy Types of the Independence and Perfect Matching Complexes of Möbius and Circular Ladder Graphs
- Equivariant Grid Homology for Strongly Invertible Knots
BibTeX
@misc{chauhan2026homology,
title = {{Homology of matching complexes of $3\times n$ grid graphs}},
author = {Pratiksha Chauhan and Anchal Sharma and Samir Shukla},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27366},
eprint = {2609.27366},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27366},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Homology of matching complexes of 3× n grid graphs
AU - Pratiksha Chauhan
AU - Anchal Sharma
AU - Samir Shukla
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27366
UR - https://arxiv.org/abs/2609.27366
AB - For a finite simple graph G, the matching complex M(G) is the simplicial complex whose vertex set is the edge set of G and whose simplices are all the matchings in G. The topology of the matching complex of the m× n grid graph G_m× n is known only for m = 1,2, in which cases it is homotopy equivalent to a wedge of spheres. In this article, we study the matching complex M(G_3 × n). We prove that for n≥2, its reduced homology vanishes in dimensions i ≤ n−2 and in top dimension, while H̃_n−1(M(G_3× n))≠ 0. We also show that M(G_3 × n) is simply connected for n ≥ 3. Consequently, the topological connectivity of M(G_3× n) is n−2.
ER -