Three Types of Two-Disjoint-Cycle-Cover Pancyclicity and Their Applications to Cycle Embedding in Locally Twisted Cubes

Abstract

Abstract A graph $G=(V,E)$ is two-disjoint-cycle-cover $[r_1,r_2]$-pancyclic if for any integer $l$ satisfying $r_1 \leq l \leq r_2$, there exist two vertex-disjoint cycles $C_1$ and $C_2$ in $G$ such that the lengths of $C_1$ and $C_2$ are $l$ and $|V(G)| - l$, respectively, where $|V(G)|$ denotes the total number of vertices in $G$. On the basis of this definition, we further propose Ore-type conditions for graphs to be two-disjoint-cycle-cover vertex/edge $[r_1,r_2]$-pancyclic. In addition, we study cycle embedding in the $n$-dimensional locally twisted cube $LTQ_n$ under the consideration of two-disjoint-cycle-cover vertex/edge pancyclicity.