Complete Cycle Embedding in Crossed Cubes with Two-Disjoint-Cycle-Cover Pancyclicity

A graph G is two-disjoint-cycle-cover r-pancyclic if for any integer l satisfying r <= 1 <= vertical bar V(G)vertical bar - r, there exist two vertex-disjoint cycles C-1 and C-2 in G such that the lengths of C-1 and C-2 are vertical bar V(G)vertical bar - l and l, respectively, where vertical bar V(G)vertical bar denotes the total number of vertices in G. In particular, the graph G is two-disjoint-cycle-cover vertex r-pancyclic if for any two distinct vertices u and u of G, there exist two vertex-disjoint cycles C-1 and C-2 in G such that (i) Ci contains u, (ii) C2 contains u, and (iii) the lengths of C-1 and C-2 are vertical bar V(G)vertical bar - 1 and 1, respectively, for any integer 1 satisfying r 1 Iv - r. Moreover, G is two-disjoint-cyclecover edge r-pancyclic if for any two vertex-disjoint edges (u, u) and (x, y) of G, there exist two vertex-disjoint cycles C-1 and C-2 in G such that (i) C-1 contains (u, u), (ii) C-2 contains (x, y), and (iii) the lengths of C1 and C-2 are vertical bar V(G)vertical bar - 1 and 1, respectively, for any integer 1 satisfying r <= 1 <= vertical bar V(G)vertical bar- r. In this paper, we first give Dirac-type sufficient conditions for general graphs to be two-disjoint-cycle-cover vertex/edge 3-pancyclic, and we also prove that the n-dimensional crossed cube CQ(n) is two-disjointcycle-cover 4-pancyclic for n >= 3, vertex 4-pancyclic for n >= 5, and edge 6-pancyclic for n >= 5.