On the Orbits of Crossed Cubes

TL;DR

This paper proves a conjecture about the number of symmetry orbits in n-dimensional crossed cubes, confirming a specific formula for all dimensions greater than or equal to three.

Contribution

It provides a proof for the conjectured formula of the orbit number of n-dimensional crossed cubes, settling a previously open problem.

Findings

Confirmed the orbit number formula for all n ≥ 3

Established the symmetry structure of crossed cubes

Validated the conjecture from prior work

Abstract

An orbit of $G$ is a subset $S$ of $V(G)$ such that $\phi(u)=v$ for any two vertices $u,v\in S$, where $\phi$ 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.