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arXiv · 2409.14015

The vertex-pancyclicity of the simplified shuffle-cube and the vertex-bipancyclicity of the balanced shuffle-cube

Yasong Liu·Huazhong Lü
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Abstract

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 $n\geqslant 2$.

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Yasong Liu, Huazhong Lü. 2024-09-21. The vertex-pancyclicity of the simplified shuffle-cube and the vertex-bipancyclicity of the balanced shuffle-cube. https://arxiv.org/abs/2409.14015

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