The minimum number of detours in a connected graph of minimum degree three
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| Format: | Preprint |
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2026
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| _version_ | 1866908995178987520 |
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| author | Liu, Xining Qiao, Pu Zhan, Xingzhi |
| author_facet | Liu, Xining Qiao, Pu Zhan, Xingzhi |
| contents | A longest path in a graph is called a detour. Denote by $a(k,n)$ the minimum number of detours in a connected graph with minimum degree $k$ and order $n,$ and denote by $b(k,n)$ the minimum odd number of detours in such a graph. X. Zhan has posed the problem of determining $a(k,n)$ and $b(k,n).$ It is known that $a(2,n)=4$ for $n\ge 4$ and $b(2,n)=9$ for $n\ge 9.$ In this paper we prove that $a(3,n)=36$ for $n\ge 18,$ $a(k,n)\le (k!)^2$ for $n\ge k^2+2k+3$ and $b(3,n)\le 225$ for $n\ge 11.$ We also pose several related unsolved problems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_24137 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The minimum number of detours in a connected graph of minimum degree three Liu, Xining Qiao, Pu Zhan, Xingzhi Combinatorics 05C30, 05C35, 05C38 A longest path in a graph is called a detour. Denote by $a(k,n)$ the minimum number of detours in a connected graph with minimum degree $k$ and order $n,$ and denote by $b(k,n)$ the minimum odd number of detours in such a graph. X. Zhan has posed the problem of determining $a(k,n)$ and $b(k,n).$ It is known that $a(2,n)=4$ for $n\ge 4$ and $b(2,n)=9$ for $n\ge 9.$ In this paper we prove that $a(3,n)=36$ for $n\ge 18,$ $a(k,n)\le (k!)^2$ for $n\ge k^2+2k+3$ and $b(3,n)\le 225$ for $n\ge 11.$ We also pose several related unsolved problems. |
| title | The minimum number of detours in a connected graph of minimum degree three |
| topic | Combinatorics 05C30, 05C35, 05C38 |
| url | https://arxiv.org/abs/2604.24137 |