Flexible cycle embedding in the locally twisted cube with nodes positioned at any prescribed distance

A Hamiltonian graph G is panpositionably Hamiltonian if for any two distinct vertices x and y of G, it contains a Hamiltonian cycle C such that 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, on C), and ∣V(G)∣ is the total number of vertices in G. As the importance of Hamiltonian properties for data communication between units in parallel and distributed systems, the panpositionable Hamiltonicity involves more flexible cycle embedding for message transmission. This paper shows that for two arbitrary nodes x and y of the n-dimensional locally twisted cube LTQn, n ⩾ 4, and for any integer l ∈ {d} ∪ {d + 2, d + 3, d + 4, … , 2n−1}, where image, there exists a Hamiltonian cycle C of LTQn such that dC(x, y) = l.