Domination numbers and homotopy in certain ternary graphs
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912515179413504 |
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| author | Eom, Taehyun Kim, Jinha Kim, Minki |
| author_facet | Eom, Taehyun Kim, Jinha Kim, Minki |
| contents | A ternary graph is a graph with no induced cycles of length $0$ modulo $3$. It was recently shown that, if the independence complex of a ternary graph is not contractible, then it is homotopy equivalent to a sphere. When a ternary graph also does not contain induced cycles of length $1$ modulo $3$, we prove that the dimension of the sphere is equal to the dimension of a minimum maximal simplex of the independence complex, or equivalently, to the value obtained by subtracting $1$ from the independent domination number of the graph. The same statement holds if we replace the independent domination number with the domination number. We also give a hypergraph analogue of the statement above. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_00699 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Domination numbers and homotopy in certain ternary graphs Eom, Taehyun Kim, Jinha Kim, Minki Combinatorics A ternary graph is a graph with no induced cycles of length $0$ modulo $3$. It was recently shown that, if the independence complex of a ternary graph is not contractible, then it is homotopy equivalent to a sphere. When a ternary graph also does not contain induced cycles of length $1$ modulo $3$, we prove that the dimension of the sphere is equal to the dimension of a minimum maximal simplex of the independence complex, or equivalently, to the value obtained by subtracting $1$ from the independent domination number of the graph. The same statement holds if we replace the independent domination number with the domination number. We also give a hypergraph analogue of the statement above. |
| title | Domination numbers and homotopy in certain ternary graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2508.00699 |