The Pairing-Hamiltonian property in graph prisms
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866913703564148736 |
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| author | Abreu, Marién Mazzuoccolo, Giuseppe Romaniello, Federico Zerafa, Jean Paul |
| author_facet | Abreu, Marién Mazzuoccolo, Giuseppe Romaniello, Federico Zerafa, Jean Paul |
| contents | Let $G$ be a graph of even order, and consider $K_G$ as the complete graph on the same vertex set as $G$. A perfect matching of $K_G$ is called a pairing of $G$. If for every pairing $M$ of $G$ it is possible to find a perfect matching $N$ of $G$ such that $M \cup N$ is a Hamiltonian cycle of $K_G$, then $G$ is said to have the Pairing-Hamiltonian property, or PH-property, for short. In 2007, Fink [J. Combin. Theory Ser. B, 97] proved that for every $d\geq 2$, the $d$-dimensional hypercube $\mathcal{Q}_d$ has the PH-property, thus proving a conjecture posed by Kreweras in 1996. In this paper we extend Fink's result by proving that given a graph $G$ having the PH-property, the prism graph $\mathcal{P}(G)$ of $G$ has the PH-property as well. Moreover, if $G$ is a connected graph, we show that there exists a positive integer $k_0$ such that the $k^{\textrm{th}}$-prism of a graph $\mathcal{P}^k(G)$ has the PH-property for all $k \ge k_0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_04545 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Pairing-Hamiltonian property in graph prisms Abreu, Marién Mazzuoccolo, Giuseppe Romaniello, Federico Zerafa, Jean Paul Combinatorics 05C76, 05C70, 05C45 Let $G$ be a graph of even order, and consider $K_G$ as the complete graph on the same vertex set as $G$. A perfect matching of $K_G$ is called a pairing of $G$. If for every pairing $M$ of $G$ it is possible to find a perfect matching $N$ of $G$ such that $M \cup N$ is a Hamiltonian cycle of $K_G$, then $G$ is said to have the Pairing-Hamiltonian property, or PH-property, for short. In 2007, Fink [J. Combin. Theory Ser. B, 97] proved that for every $d\geq 2$, the $d$-dimensional hypercube $\mathcal{Q}_d$ has the PH-property, thus proving a conjecture posed by Kreweras in 1996. In this paper we extend Fink's result by proving that given a graph $G$ having the PH-property, the prism graph $\mathcal{P}(G)$ of $G$ has the PH-property as well. Moreover, if $G$ is a connected graph, we show that there exists a positive integer $k_0$ such that the $k^{\textrm{th}}$-prism of a graph $\mathcal{P}^k(G)$ has the PH-property for all $k \ge k_0$. |
| title | The Pairing-Hamiltonian property in graph prisms |
| topic | Combinatorics 05C76, 05C70, 05C45 |
| url | https://arxiv.org/abs/2307.04545 |