The minimum size of a $3$-connected locally nonforesty graph
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910679035805696 |
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| author | Li, Chengli Tang, Yurui Zhan, Xingzhi |
| author_facet | Li, Chengli Tang, Yurui Zhan, Xingzhi |
| contents | A local subgraph of a graph is the subgraph induced by the neighborhood of a vertex. Thus a graph of order $n$ has $n$ local subgraphs. A graph $G$ is called locally nonforesty if every local subgraph of $G$ contains a cycle. Recently, in studying forest cuts of a graph, Chernyshev, Rauch and Rautenbach posed the conjecture that if $n$ and $m$ are the order and size of a $3$-connected locally nonforesty graph respectively, then $m\ge 7(n-1)/3.$ We solve this problem by determining the minimum size of a $3$-connected locally nonforesty graph of order $n.$ It turns out that the conjecture does not hold. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_23702 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The minimum size of a $3$-connected locally nonforesty graph Li, Chengli Tang, Yurui Zhan, Xingzhi Combinatorics 05C35, 05C38, 05C40 A local subgraph of a graph is the subgraph induced by the neighborhood of a vertex. Thus a graph of order $n$ has $n$ local subgraphs. A graph $G$ is called locally nonforesty if every local subgraph of $G$ contains a cycle. Recently, in studying forest cuts of a graph, Chernyshev, Rauch and Rautenbach posed the conjecture that if $n$ and $m$ are the order and size of a $3$-connected locally nonforesty graph respectively, then $m\ge 7(n-1)/3.$ We solve this problem by determining the minimum size of a $3$-connected locally nonforesty graph of order $n.$ It turns out that the conjecture does not hold. |
| title | The minimum size of a $3$-connected locally nonforesty graph |
| topic | Combinatorics 05C35, 05C38, 05C40 |
| url | https://arxiv.org/abs/2410.23702 |