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

Abstract

A graph G is two-disjoint-cycle-cover r-pancyclic if for any integer l satisfying r≤l≤|V(G)|-r, there exist two vertex-disjoint cycles C₁ and C₂ in G such that the lengths of C₁ and C₂ are |V(G)|-l and l, respectively, where |V(G)| 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 v of G, there exist two vertex-disjoint cycles C₁ and C₂ in G such that (i) C₁ contains u, (ii) C₂ contains v, and (iii) the lengths of C₁ and C₂ are |V(G)|-l and l, respectively, for any integer l satisfying r≤l≤|V(G)|-r. Moreover, G is two-disjoint-cycle-cover edge r-pancyclic if for any two vertex-disjoint edges (u,v) and (x,y) of G, there exist two vertex-disjoint cycles C₁ and C₂ in G such that (i) C₁ contains (u,v), (ii) C₂ contains (x,y), and (iii) the lengths of C₁ and C₂ are |V(G)|-l and l, respectively, for any integer l satisfying r≤l≤|V(G)|-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-disjoint-cycle-cover 4-pancyclic for n≥3, vertex 4-pancyclic for n≥5, and edge 6-pancyclic for n≥5.