Hamiltonian paths extending a set of matchings in hypercubes
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915360114999296 |
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| author | Ali, Abid Ba, Lina Yang, Weihua |
| author_facet | Ali, Abid Ba, Lina Yang, Weihua |
| contents | The hypercube \( Q_n \) contains a Hamiltonian path joining \( x \) and \( y \) (where $x$ and $y$ from the opposite partite set) containing \( P \) if and only if the induced subgraph of \( P \) is a linear forest, where none of these paths have \( x \) or \( y \) as internal vertices nor both as endpoints. Dvořák and Gregor answered a problem posed by Caha and Koubek and proved that for every \( n \geq 5 \), there exist vertices \( x \) and \( y \) with a set of \( 2n - 4 \) edges in \( Q_n \) that extend to the Hamiltonian path joining \( x \) and \( y \). This paper examines the Hamiltonian properties of hypercubes with a matching set. Let consider the hypercube \( Q_n \), for \( n \geq 5 \) and a set of matching \( M \) such that \( |M| \leq 3n - 13 \). We prove a Hamiltonian path exists joining two vertices $x$ and $y$ in \( Q_n \) from opposite partite sets containing $M$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_21432 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hamiltonian paths extending a set of matchings in hypercubes Ali, Abid Ba, Lina Yang, Weihua Combinatorics 05C70, 05C45 F.2.2 The hypercube \( Q_n \) contains a Hamiltonian path joining \( x \) and \( y \) (where $x$ and $y$ from the opposite partite set) containing \( P \) if and only if the induced subgraph of \( P \) is a linear forest, where none of these paths have \( x \) or \( y \) as internal vertices nor both as endpoints. Dvořák and Gregor answered a problem posed by Caha and Koubek and proved that for every \( n \geq 5 \), there exist vertices \( x \) and \( y \) with a set of \( 2n - 4 \) edges in \( Q_n \) that extend to the Hamiltonian path joining \( x \) and \( y \). This paper examines the Hamiltonian properties of hypercubes with a matching set. Let consider the hypercube \( Q_n \), for \( n \geq 5 \) and a set of matching \( M \) such that \( |M| \leq 3n - 13 \). We prove a Hamiltonian path exists joining two vertices $x$ and $y$ in \( Q_n \) from opposite partite sets containing $M$. |
| title | Hamiltonian paths extending a set of matchings in hypercubes |
| topic | Combinatorics 05C70, 05C45 F.2.2 |
| url | https://arxiv.org/abs/2506.21432 |