Wedge-of-spheres structure of path-product matching complexes
Let $P_n$ be the path graph on $\{1, \dots, n\}$ and let $G_{n,m} = P_n \times P_m$ be the categorical product of $P_n$ and $P_m$: its vertices are the pairs $(i, j)$, and $(i, j)$ is joined to $(i', j')$ exactly when $|i - i'| = |j - j'| = 1$. For a graph $G$, let $\mathsf{M}(G)$ be its matching complex and $|\mathsf{M}(G)|$ the geometric realization. The graph $P_n \times P_2$ is a disjoint union of two paths, so its matching complex is homotopy equivalent to a wedge of spheres by Kozlov's computation for paths. Gupta, Sarkar, Sawant and Shukla proved that $\mathsf{M}(P_n \times P_m)$ is homotopy equivalent to a wedge of spheres for $n \ge 2$ and $3 \le m \le 5$; for $m = 3$ they determine the number and dimensions of the spheres, and for $m = 4, 5$ the least and greatest dimensions. Since $G_{n,m} \cong G_{m,n}$, this covers every pair in which one of $n, m$ lies between $2$ and $5$.
Problem. For every pair of integers $n, m \ge 6$, is $$|\mathsf{M}(G_{n,m})| \simeq \bigvee_{a=1}^{r} S^{d_a}$$ for some finite list of integers $d_1, \dots, d_r \ge 0$, where $\simeq$ is homotopy equivalence, the spheres may have different dimensions, and $r = 0$ stands for a point?
Source: Gupta, Sarkar, Sawant and Shukla (2026), Conjecture 6.2, for n ≥ 3 and m ≥ 6 (arXiv 2024). Stated by AIM — Open Applied Problems for n, m ≥ 6, the range their theorem leaves.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- CO — Combinatorics
- Secondary
- AT
- Keywords
- AIM Open Applied Problems, categorical products of graphs, matching complexes, wedges of spheres
References
@article{gupta2026,
author = {Gupta, Raju Kumar and Sarkar, Sourav and Sawant, Sagar S. and Shukla, Samir},
title = {On the matching complexes of the categorical product of path graphs},
journal = {J. Appl. Comput. Topol.},
volume = {10},
number = {3},
year = {2026},
pages = {19},
doi = {10.1007/s41468-026-00238-y},
eprint = {2403.15298},
archivePrefix = {arXiv},
}
@article{kozlov1999,
author = {Kozlov, Dmitry N.},
title = {Complexes of directed trees},
journal = {J. Combin. Theory Ser. A},
volume = {88},
number = {1},
year = {1999},
pages = {112--122},
doi = {10.1006/jcta.1999.2984},
}
@misc{aim-path-product-matching-complexes,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Wedge-of-spheres structure of path-product matching complexes}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/509-path-product-matching-complexes.md},
note = {Entry 509 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00885, "Wedge-of-spheres structure of path-product matching complexes", clopen.solutions.