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

Abstract 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 d C ( x ,  y ) =  l for any integer l satisfying d G ( x ,  y ) ⩽  l  ⩽ ⌈∣ V ( G )∣/2⌉, where d G ( x ,  y ) (respectively, d C ( 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 LTQ n , n  ⩾ 4, and for any integer l  ∈ { d } ∪ { d  + 2, d  + 3, d  + 4, … , 2 n −1 }, where d = d LTQ n ( x , y ) , there exists a Hamiltonian cycle C of LTQ n such that d C ( x ,  y ) =  l .