Two-Disjoint-Cycle-Cover Pancyclicity of (n, k)-Star Graph

Let G be a connected graph. Let $$t_1\geqslant 3$$ , $$t_2\geqslant 3$$ be two integers with $$t_1\leqslant t_2$$ . Then G is two-disjoint-cycle-cover $$[t_1,t_2]$$ -pancyclic or briefly 2-DCC $$[t_1,t_2]$$ -pancyclic if for any integer t with $$t_1\leqslant t\leqslant t_2$$ , G has two cycles $$C_1$$ and $$C_2$$ satisfying $$\vert V(C_1)\vert =t$$ and $$V(C_2)=V(G)-V(C_1)$$ . Let $$q,w\in V(G)$$ with $$q\ne w$$ be two arbitrary vertices. Then G is 2-DCC vertex $$[t_1,t_2]$$ -pancyclic if for any integer t with $$t_1\leqslant t\leqslant t_2$$ , G has two cycles $$C_1,C_2$$ such that $$q\in V(C_1), w\in V(C_2)$$ , where $$\vert V(C_1)\vert =t$$ , $$V(C_2)=V(G)-V(C_1)$$ . We show that (n, 1)-star graph $$S_{n,1}$$ is 2-DCC edge $$[3,\lfloor \frac{n}{2}\rfloor ]$$ -pancyclic, 2-DCC vertex $$[3,\lfloor \frac{n}{2}\rfloor ]$$ -pancyclic and 2-DCC $$[3,\lfloor \frac{n}{2}\rfloor ]$$ -pancyclic when $$n\geqslant 6$$ . We obtain that (n, k)-star graph $$S_{n,k}$$ is 2-DCC $$[3, \frac{n!}{2(n-k)!}]$$ -pancyclic when $$n\geqslant 8$$ and $$2 \leqslant k \leqslant n-6.$$