Saved by the rook: a case of matchings and Hamiltonian cycles
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866911039874924544 |
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| author | Abreu, Marién Gauci, John Baptist Zerafa, Jean Paul |
| author_facet | Abreu, Marién Gauci, John Baptist Zerafa, Jean Paul |
| contents | The rook graph is a graph whose edges represent all the possible legal moves of the rook chess piece on a chessboard. The problem we consider is the following. Given any set $M$ containing pairs of cells such that each cell of the $m_1 \times m_2$ chessboard is in exactly one pair, we determine the values of the positive integers $m_1$ and $m_2$ for which it is possible to construct a closed tour of all the cells of the chessboard which uses all the pairs of cells in $M$ and some edges of the rook graph. This is an alternative formulation of a graph-theoretical problem presented in [Electron. J. Combin. 28(1) (2021), #P1.7] involving the Cartesian product $G$ of two complete graphs $K_{m_1}$ and $K_{m_2}$, which is, in fact, isomorphic to the $m_{1}\times m_{2}$ rook graph. The problem revolves around determining the values of the parameters $m_1$ and $m_2$ that would allow any perfect matching of the complete graph on the same vertex set of $G$ to be extended to a Hamiltonian cycle by using only edges in $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_01578 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Saved by the rook: a case of matchings and Hamiltonian cycles Abreu, Marién Gauci, John Baptist Zerafa, Jean Paul Combinatorics 05C45, 05C70, 05C76 The rook graph is a graph whose edges represent all the possible legal moves of the rook chess piece on a chessboard. The problem we consider is the following. Given any set $M$ containing pairs of cells such that each cell of the $m_1 \times m_2$ chessboard is in exactly one pair, we determine the values of the positive integers $m_1$ and $m_2$ for which it is possible to construct a closed tour of all the cells of the chessboard which uses all the pairs of cells in $M$ and some edges of the rook graph. This is an alternative formulation of a graph-theoretical problem presented in [Electron. J. Combin. 28(1) (2021), #P1.7] involving the Cartesian product $G$ of two complete graphs $K_{m_1}$ and $K_{m_2}$, which is, in fact, isomorphic to the $m_{1}\times m_{2}$ rook graph. The problem revolves around determining the values of the parameters $m_1$ and $m_2$ that would allow any perfect matching of the complete graph on the same vertex set of $G$ to be extended to a Hamiltonian cycle by using only edges in $G$. |
| title | Saved by the rook: a case of matchings and Hamiltonian cycles |
| topic | Combinatorics 05C45, 05C70, 05C76 |
| url | https://arxiv.org/abs/2104.01578 |