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

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.