The vertex-pancyclicity of the simplified shuffle-cube and the vertex-bipancyclicity of the balanced shuffle-cube
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914954483859456 |
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| author | Liu, Yasong Lü, Huazhong |
| author_facet | Liu, Yasong Lü, Huazhong |
| contents | A graph $G$ $=$ $(V,E)$ is vertex-pancyclic if for every vertex $u$ and any integer $l$ ranging from $3$ to $|V|$, $G$ contains a cycle $C$ of length $l$ such that $u$ is on $C$. A bipartite graph $G$ $=$ $(V,E)$ is vertex-bipancyclic if for every vertex $u$ and any even integer $l$ ranging from $4$ to $|V|$, $G$ contains a cycle $C$ of length $l$ such that $u$ is on $C$. The simplified shuffle-cube and the balanced shuffle-cube, which are two variants of the shuffle-cube and are superior to shuffle-cube in terms of vertex-transitivity. In this paper, we show that the $n$-dimensional simplified shuffle-cube is vertex-pancyclic for $n\geqslant 6$, and the $n$-dimensional balanced shuffle-cube is vertex-bipancyclic for $n\geqslant 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_14015 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The vertex-pancyclicity of the simplified shuffle-cube and the vertex-bipancyclicity of the balanced shuffle-cube Liu, Yasong Lü, Huazhong Combinatorics Discrete Mathematics A graph $G$ $=$ $(V,E)$ is vertex-pancyclic if for every vertex $u$ and any integer $l$ ranging from $3$ to $|V|$, $G$ contains a cycle $C$ of length $l$ such that $u$ is on $C$. A bipartite graph $G$ $=$ $(V,E)$ is vertex-bipancyclic if for every vertex $u$ and any even integer $l$ ranging from $4$ to $|V|$, $G$ contains a cycle $C$ of length $l$ such that $u$ is on $C$. The simplified shuffle-cube and the balanced shuffle-cube, which are two variants of the shuffle-cube and are superior to shuffle-cube in terms of vertex-transitivity. In this paper, we show that the $n$-dimensional simplified shuffle-cube is vertex-pancyclic for $n\geqslant 6$, and the $n$-dimensional balanced shuffle-cube is vertex-bipancyclic for $n\geqslant 2$. |
| title | The vertex-pancyclicity of the simplified shuffle-cube and the vertex-bipancyclicity of the balanced shuffle-cube |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2409.14015 |