On decycling and forest numbers of Cartesian products of trees
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914188283084800 |
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| author | Ghalavand, Ali Klavžar, Sandi Yang, Ning |
| author_facet | Ghalavand, Ali Klavžar, Sandi Yang, Ning |
| contents | The decycling number $\nabla(G)$ of a graph $G$ is the minimum number of vertices that must be removed to eliminate all cycles in $G$. The forest number $f(G)$ is the maximum number of vertices that induce a forest in $G$. So $\nabla(G) + f(G) = |V(G)|$. For the Cartesian product $T \,\square\, T'$ of trees $T$ and $T'$ it is proved that $\nabla(S_n \,\square\, S_{n'}) \leq \nabla(T \,\square\, T')$, thus resolving the conjecture of Wang and Wu asserting that $f(T \,\square\, T') \leq f(S_n \,\square\, S_{n'})$. It is shown that $\nabla(T \,\square\, T') \ge |V(T)| - 1$ and the equality cases characterized. For prisms over trees, it is proved that $\nabla(T\,\square\, K_2) = α'(T)$, and for arbitrary graphs $G_1$ and $G_2$, it is proved that $\nabla(G_1 \,\square\, G_2) \geq α'(G_1) α'(G_2)$, where $α'$ is the matching number. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_06902 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On decycling and forest numbers of Cartesian products of trees Ghalavand, Ali Klavžar, Sandi Yang, Ning Combinatorics The decycling number $\nabla(G)$ of a graph $G$ is the minimum number of vertices that must be removed to eliminate all cycles in $G$. The forest number $f(G)$ is the maximum number of vertices that induce a forest in $G$. So $\nabla(G) + f(G) = |V(G)|$. For the Cartesian product $T \,\square\, T'$ of trees $T$ and $T'$ it is proved that $\nabla(S_n \,\square\, S_{n'}) \leq \nabla(T \,\square\, T')$, thus resolving the conjecture of Wang and Wu asserting that $f(T \,\square\, T') \leq f(S_n \,\square\, S_{n'})$. It is shown that $\nabla(T \,\square\, T') \ge |V(T)| - 1$ and the equality cases characterized. For prisms over trees, it is proved that $\nabla(T\,\square\, K_2) = α'(T)$, and for arbitrary graphs $G_1$ and $G_2$, it is proved that $\nabla(G_1 \,\square\, G_2) \geq α'(G_1) α'(G_2)$, where $α'$ is the matching number. |
| title | On decycling and forest numbers of Cartesian products of trees |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2501.06902 |