Embedding a Hamiltonian cycle in the crossed cube with two required vertices in the fixed positions

A Hamiltonian graph G is said to be panpositionably Hamiltonian if, for any two distinct vertices x and y of G, there is a Hamiltonian cycle C of G having dC(x, y) = l for any integer l   satisfying dG(x,y)⩽l⩽|V(G)|2, where dG(x, y) (respectively, dC(x, y)) denotes the distance between vertices x and y in G (respectively, C), and ∣V(G)∣ denotes the total number of vertices of G. As the importance of Hamiltonian properties for data communication among units in an interconnected system, the panpositionable Hamiltonicity involves more flexible message transmission. In this paper, we study this property with respect to the class of crossed cubes, which is a popular variant of the hypercube network.