The maximum number of paths of a given length in a nonhamiltonian graph
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866916046886141952 |
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| author | Li, Chengli Zhan, Xingzhi |
| author_facet | Li, Chengli Zhan, Xingzhi |
| contents | In 1980, Paul Erdős posed the following problem: For every positive integer $n,$ determine a nonhamiltonian graph of order $n$ having the maximum number of Hamilton paths. We solve the more general problem of determining the nonhamiltonian graphs of order $n$ having the maximum number of paths of length $k$ for given integers $n$ and $k$ with $1\le k\le n-1.$ The case $k=n-1$ gives a solution to Erdős's problem and the case $k=1$ corresponds to a theorem due to Ore and Bondy. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_26479 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The maximum number of paths of a given length in a nonhamiltonian graph Li, Chengli Zhan, Xingzhi Combinatorics 05C30, 05C35, 05C38 In 1980, Paul Erdős posed the following problem: For every positive integer $n,$ determine a nonhamiltonian graph of order $n$ having the maximum number of Hamilton paths. We solve the more general problem of determining the nonhamiltonian graphs of order $n$ having the maximum number of paths of length $k$ for given integers $n$ and $k$ with $1\le k\le n-1.$ The case $k=n-1$ gives a solution to Erdős's problem and the case $k=1$ corresponds to a theorem due to Ore and Bondy. |
| title | The maximum number of paths of a given length in a nonhamiltonian graph |
| topic | Combinatorics 05C30, 05C35, 05C38 |
| url | https://arxiv.org/abs/2605.26479 |