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| Format: | Preprint |
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2024
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| Online-Zugang: | https://arxiv.org/abs/2403.15298 |
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| _version_ | 1866915951160590336 |
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| author | Gupta, Raju Kumar Sarkar, Sourav Sawant, Sagar S. Shukla, Samir |
| author_facet | Gupta, Raju Kumar Sarkar, Sourav Sawant, Sagar S. Shukla, Samir |
| contents | The matching complex $\mathsf{M}(G)$ of a graph $G$ is a simplicial complex whose simplices are matchings in $G$. These complexes appear in various places and found applications in many areas of mathematics including computational geometry, representation theory, combinatorics, etc.
In this article, we consider the matching complexes of the categorical product $P_n \times P_m$ of path graphs $P_n$ and $P_m$. For $m = 1$, $P_n \times P_m$ is a discrete graph and therefore its matching complex is the void complex. For $m = 2$, $\M(P_n \times P_m)$ has been proved to be homotopy equivalent to a wedge of spheres by Kozlov. We show that for $n \geq 2$ and $3 \leq m \leq 5$, the matching complex of $P_n \times P_m$ is homotopy equivalent to a wedge of spheres. For $m =3$, we explicitly compute the number and dimension of spheres appearing in the wedge. Furthermore, for $m \in \{4, 5\}$, we provide the minimum and maximum dimensions of spheres appearing in the wedge in the homotopy type of $\mathsf{M}(P_n \times P_m)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_15298 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the matching complexes of categorical product of path graphs Gupta, Raju Kumar Sarkar, Sourav Sawant, Sagar S. Shukla, Samir Combinatorics 55P10, 05E45, 55U10 The matching complex $\mathsf{M}(G)$ of a graph $G$ is a simplicial complex whose simplices are matchings in $G$. These complexes appear in various places and found applications in many areas of mathematics including computational geometry, representation theory, combinatorics, etc. In this article, we consider the matching complexes of the categorical product $P_n \times P_m$ of path graphs $P_n$ and $P_m$. For $m = 1$, $P_n \times P_m$ is a discrete graph and therefore its matching complex is the void complex. For $m = 2$, $\M(P_n \times P_m)$ has been proved to be homotopy equivalent to a wedge of spheres by Kozlov. We show that for $n \geq 2$ and $3 \leq m \leq 5$, the matching complex of $P_n \times P_m$ is homotopy equivalent to a wedge of spheres. For $m =3$, we explicitly compute the number and dimension of spheres appearing in the wedge. Furthermore, for $m \in \{4, 5\}$, we provide the minimum and maximum dimensions of spheres appearing in the wedge in the homotopy type of $\mathsf{M}(P_n \times P_m)$. |
| title | On the matching complexes of categorical product of path graphs |
| topic | Combinatorics 55P10, 05E45, 55U10 |
| url | https://arxiv.org/abs/2403.15298 |