Two-disjoint-cycle-cover edge/vertex bipancyclicity of star graphs

Abstract

A bipartite graph $G$ is two-disjoint-cycle-cover edge $[r_1,r_2]$-bipancyclic if, for any vertex-disjoint edges $uv$ and $xy$ in $G$ and any even integer $\ell$ satisfying $r_1⩽\ell ⩽r_2$, there exist vertex-disjoint cycles $C_1$ and $C_2$ such that $\left|V\left(C_1\right)\right|=\ell$, $\left|V\left(C_2\right)\right|=\left|V\left(G\right)\right|-\ell$, $uv\in E\left(C_1\right)$ and $xy\in E\left(C_2\right)$. In this paper, we prove that the $n$-star graph $S_n$ is two-disjoint-cycle-cover edge $[6,\frac{n!}{2}]$-bipancyclic for $n⩾5$, and thus it is two-disjoint-cycle-cover vertex $[6,\frac{n!}{2}]$-bipancyclic for $n⩾5$. Additionally, it is examined that $S_n$ is two-disjoint-cycle-cover $[6,\frac{n!}{2}]$-bipancyclic for $n⩾4$.