Two-disjoint-cycle-cover pancyclicity of augmented cubes

Abstract

A graph $G$ is two-disjoint-cycle-cover $[r_1,r_2]$-pancyclic if there is a collection of two vertex disjoint cycles $C$ of length $l\left(C\right)$ and $C^{\prime }$ of length $l\left(C^{\prime }\right)$ such that $l\left(C\right)+l\left(C^{\prime }\right)=\left|V\left(G\right)\right|$ and $r_1\le l\left(C\right)\le r_2$. The augmented cube is one of the variations of hypercube and possesses many good properties that the hypercube does not have. In this paper, we prove that the $n$-dimensional augmented cube $A{Q}_n$ is two-disjoint-cycle-cover $[3,2^{n-1}]$-pancyclic, where $n\ge 3$.