Symmetric property and edge-disjoint Hamiltonian cycles of the spined cube

TL;DR

This paper investigates the symmetric properties of spined cubes, showing they can be decomposed into hypercubes and contain edge-disjoint Hamiltonian cycles for certain dimensions, with implications for network design.

Contribution

It proves that spined cubes are multi-Cayley graphs with specific symmetry properties and establishes the existence of edge-disjoint Hamiltonian cycles in these graphs.

Findings

$SQ_n$ is a 4-Cayley graph for $n\,\geq6$

$SQ_n$ contains two edge-disjoint Hamiltonian cycles for $n\geq4$

$SQ_n$ is not vertex-transitive unless $n\leq3$

Abstract

The spined cube $SQ_n$ is a variant of the hypercube $Q_n$, introduced by Zhou et al. in [Information Processing Letters 111 (2011) 561-567] as an interconnection network for parallel computing. A graph $\G$ is an $m$-Cayley graph if its automorphism group $\Aut(\G)$ has a semiregular subgroup acting on the vertex set with $m$ orbits, and is a Caley graph if it is a 1-Cayley graph. It is well-known that $Q_n$ is a Cayley graph of an elementary abelian 2-group $\mz_2^n$ of order $2^n$. In this paper, we prove that $SQ_n$ is a 4-Cayley graph of $\mz_2^{n-2}$ when $n\geq6$, and is a $\lfloor n/2\rfloor$-Cayley graph when $n\leq 5$. This symmetric property shows that an $n$-dimensional spined cube with $n\geq6$ can be decomposed to eight vertex-disjoint $(n-3)$-dimensional hypercubes, and as an application, it is proved that there exist two edge-disjoint Hamiltonian cycles in $SQ_n$ when $n\geq4$. Moreover, we determine the vertex-transitivity of $SQ_n$, and prove that $SQ_n$ is not vertex-transitive unless $n\leq3$.