Two-disjoint-cycle-cover vertex pancyclicity of split-star networks

Let $r_1$ and $r_2$ be positive integers with $r_1\le r_2$. A graph $G$ is called $2$-DCC vertex $[r_1,r_2]$-pancyclic if, for any two distinct vertices of $G$ and any integer $\mathrm{\ell }\in [r_1,r_2]$, there exist two vertex-disjoint cycles of lengths $\mathrm{\ell }$ and $\mathrm{\mid }V(G)\mathrm{\mid }-\mathrm{\ell }$, respectively, containing the two vertices separately. In this paper, we investigate the two-disjoint-cycle-cover vertex pancyclicity of the split-star network $S_n^2$. We prove that $S_n^2$ is $2$-DCC vertex $[3,n!\mathrm{/}2]$-pancyclic for $n\ge 4$.