Homotopy type of the independence complex of some categorical products of graphs
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866916864141033472 |
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| author | Camarena, Omar Antolín Bravo, Andrés Carnero |
| author_facet | Camarena, Omar Antolín Bravo, Andrés Carnero |
| contents | It was conjectured by Goyal, Shukla and Singh that the independence complex of the categorical product $K_2\times K_3\times K_n$ has the homotopy type of a wedge of $(n-1)(3n-2)$ spheres of dimension $3$. Here we prove this conjecture by calculating the homotopy type of the independence complex of the graphs $C_{3r}\times K_n$ and $K_2\times K_m\times K_n$. For $C_m \times K_n$ when $m$ is not a multiple of $3$, we calculate the homotopy type for $m = 4, 5$ and show that for other values it has to have the homotopy type of a wedge of spheres of at most $2$ consecutive dimensions and maybe some Moore spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_12401 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Homotopy type of the independence complex of some categorical products of graphs Camarena, Omar Antolín Bravo, Andrés Carnero Algebraic Topology Combinatorics It was conjectured by Goyal, Shukla and Singh that the independence complex of the categorical product $K_2\times K_3\times K_n$ has the homotopy type of a wedge of $(n-1)(3n-2)$ spheres of dimension $3$. Here we prove this conjecture by calculating the homotopy type of the independence complex of the graphs $C_{3r}\times K_n$ and $K_2\times K_m\times K_n$. For $C_m \times K_n$ when $m$ is not a multiple of $3$, we calculate the homotopy type for $m = 4, 5$ and show that for other values it has to have the homotopy type of a wedge of spheres of at most $2$ consecutive dimensions and maybe some Moore spaces. |
| title | Homotopy type of the independence complex of some categorical products of graphs |
| topic | Algebraic Topology Combinatorics |
| url | https://arxiv.org/abs/2307.12401 |