On the number of Hamiltonian cycles in the generalized Petersen graph
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913914609991680 |
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| author | Haugland, Jan Kristian |
| author_facet | Haugland, Jan Kristian |
| contents | The generalized Petersen graph $G(n, k)$ is a cubic graph with vertex set $V(G(n, k)) = \{v_i\}_{0 \leq i < n} \cup \{w_i\}_{0 \leq i < n}$ and edge set $E(G(n, k)) = \{v_i v_{i+1}\}_{0 \leq i < n} \cup \{w_i w_{i+k}\}_{0 \leq i < n} \cup \{v_i w_i\}_{0 \leq i < n}$ where the indices are taken modulo $n$. Schwenk found the number of Hamiltonian cycles in $G(n, 2)$, and in this article we present initial conditions and linear recurrence relations for the number of Hamiltonian cycles in $G(n, 3)$ and $G(n, 4)$. This is attained by introducing $G'(n, k)$, which is a modified version of $G(n, k)$, and a subset of its subgraphs which we call admissible, and which are partitioned into different classes in such a manner that we can find relations between the number of admissible subgraphs of each class. The classes and their relations define a directed graph such that each strongly connected component is of a manageable size for $k=3$ and $k=4$, which allows us to find linear recurrence relations for the number of admissible subgraphs in each class in these cases. The number of Hamiltonian cycles in $G(n, k)$ is a sum of the number of admissible subgraphs of $G'(n, k)$ over a certain subset of the classes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_08326 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the number of Hamiltonian cycles in the generalized Petersen graph Haugland, Jan Kristian Combinatorics 05C30 G.2.2 The generalized Petersen graph $G(n, k)$ is a cubic graph with vertex set $V(G(n, k)) = \{v_i\}_{0 \leq i < n} \cup \{w_i\}_{0 \leq i < n}$ and edge set $E(G(n, k)) = \{v_i v_{i+1}\}_{0 \leq i < n} \cup \{w_i w_{i+k}\}_{0 \leq i < n} \cup \{v_i w_i\}_{0 \leq i < n}$ where the indices are taken modulo $n$. Schwenk found the number of Hamiltonian cycles in $G(n, 2)$, and in this article we present initial conditions and linear recurrence relations for the number of Hamiltonian cycles in $G(n, 3)$ and $G(n, 4)$. This is attained by introducing $G'(n, k)$, which is a modified version of $G(n, k)$, and a subset of its subgraphs which we call admissible, and which are partitioned into different classes in such a manner that we can find relations between the number of admissible subgraphs of each class. The classes and their relations define a directed graph such that each strongly connected component is of a manageable size for $k=3$ and $k=4$, which allows us to find linear recurrence relations for the number of admissible subgraphs in each class in these cases. The number of Hamiltonian cycles in $G(n, k)$ is a sum of the number of admissible subgraphs of $G'(n, k)$ over a certain subset of the classes. |
| title | On the number of Hamiltonian cycles in the generalized Petersen graph |
| topic | Combinatorics 05C30 G.2.2 |
| url | https://arxiv.org/abs/2503.08326 |