Small hitting sets for longest paths and cycles
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913973922693120 |
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| author | Norin, Sergey Steiner, Raphael Thomassé, Stephan Wollan, Paul |
| author_facet | Norin, Sergey Steiner, Raphael Thomassé, Stephan Wollan, Paul |
| contents | Motivated by an old question of Gallai (1966) on the intersection of longest paths in a graph and the well-known conjectures of Lovász (1969) and Thomassen (1978) on the maximum length of paths and cycles in vertex-transitive graphs, we present improved bounds for the parameters $\mathrm{lpt}(G)$ and $\mathrm{lct}(G)$, defined as the minimum size of a set of vertices in a graph $G$ hitting all longest paths (cycles, respectively). First, we show that every connected graph $G$ on $n$ vertices satisfies $\mathrm{lpt}(G)\le \sqrt{8n}$, and $\mathrm{lct}(G)\le \sqrt{8n}$ if $G$ is additionally $2$-connected. This improves a sequence of earlier bounds for these problems, with the previous state of the art being $O(n^{2/3})$. Second, we show that every connected graph $G$ satisfies $\mathrm{lpt}(G)\le O(\ell^{5/9})$, where $\ell$ denotes the maximum length of a path in $G$. As an immediate application of this latter bound, we present further progress towards Lovász' and Thomassen's conjectures: We show that every connected vertex-transitive graph of order $n$ contains a cycle (and path) of length $Ω(n^{9/14})$. This improves the previous best bound of the form $Ω(n^{13/21})$. Interestingly, our proofs make use of several concepts and results from structural graph theory, such as a result of Robertson and Seymour (1990) on transactions in societies and Tutte's $2$-separator theorem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_08634 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Small hitting sets for longest paths and cycles Norin, Sergey Steiner, Raphael Thomassé, Stephan Wollan, Paul Combinatorics 05C38, 05C69 Motivated by an old question of Gallai (1966) on the intersection of longest paths in a graph and the well-known conjectures of Lovász (1969) and Thomassen (1978) on the maximum length of paths and cycles in vertex-transitive graphs, we present improved bounds for the parameters $\mathrm{lpt}(G)$ and $\mathrm{lct}(G)$, defined as the minimum size of a set of vertices in a graph $G$ hitting all longest paths (cycles, respectively). First, we show that every connected graph $G$ on $n$ vertices satisfies $\mathrm{lpt}(G)\le \sqrt{8n}$, and $\mathrm{lct}(G)\le \sqrt{8n}$ if $G$ is additionally $2$-connected. This improves a sequence of earlier bounds for these problems, with the previous state of the art being $O(n^{2/3})$. Second, we show that every connected graph $G$ satisfies $\mathrm{lpt}(G)\le O(\ell^{5/9})$, where $\ell$ denotes the maximum length of a path in $G$. As an immediate application of this latter bound, we present further progress towards Lovász' and Thomassen's conjectures: We show that every connected vertex-transitive graph of order $n$ contains a cycle (and path) of length $Ω(n^{9/14})$. This improves the previous best bound of the form $Ω(n^{13/21})$. Interestingly, our proofs make use of several concepts and results from structural graph theory, such as a result of Robertson and Seymour (1990) on transactions in societies and Tutte's $2$-separator theorem. |
| title | Small hitting sets for longest paths and cycles |
| topic | Combinatorics 05C38, 05C69 |
| url | https://arxiv.org/abs/2505.08634 |