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arXiv · 1707.06763

On the Orbits of Crossed Cubes

Tzong-Huei Shiau·Yue-Li Wang·Kung-Jui Pai
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Abstract

An orbit of $G$ is a subset $S$ of $V(G)$ such that $ϕ(u)=v$ for any two vertices $u,v\in S$, where $ϕ$ is an isomorphism of $G$. The orbit number of a graph $G$, denoted by $\text{Orb}(G)$, is the number of orbits of $G$. In [A Note on Path Embedding in Crossed Cubes with Faulty Vertices, Information Processing Letters 121 (2017) pp. 34--38], Chen et al. conjectured that $\text{Orb}(\text{CQ}_n)=2^{\lceil\frac{n}{2}\rceil-2}$ for $n\geqslant 3$, where $\text{CQ}_n$ denotes an $n$-dimensional crossed cube. In this paper, we settle the conjecture.

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Tzong-Huei Shiau, Yue-Li Wang, Kung-Jui Pai. 2017-07-31. On the Orbits of Crossed Cubes. https://arxiv.org/abs/1707.06763

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