The Panpositionable Pancyclicity of Locally Twisted Cubes

Abstract

In a multiprocessor system, processors are connected based on various types of network topologies. A network topology is usually represented by a graph. Let G be a graph and u, v be any two distinct vertices of G. We say that G is pancyclic if G has a cycle C of every length l(C) satisfying 3≤l(C)≤|V(G)|, where |V(G)| denotes the total number of vertices in G. Moreover, G is panpositionably pancyclic from r if for any integer m satisfying $r \leq m \leq \frac{|V(G)|}{2}$, G has a cycle C containing u and v such that d_C(u,v)=m and 2m≤l(C)≤|V(G)|, where d_C(u,v) denotes the distance of u and v in C. In this paper, we investigate the panpositionable pancyclicity problem with respect to the n-dimensional locally twisted cube LTQ_n, which is a popular topology derived from the hypercube. Let D(LTQ_n) denote the diameter of LTQ_n. We show that for n≥4 and for any integer m satisfying $D(LTQ_n) + 2 \leq m \leq \frac{|V(LTQ_n)|}{2}$, there exists a cycle C of LTQ_n such that d_C(u,v)=m, where (i) 2m+1≤l(C)≤|V(LTQ_n)| if m=D(LTQ_n)+2 and n is odd, and (ii) 2m≤l(C)≤|V(LTQ_n)| otherwise. This improves on the recent result that u and v can be positioned with a given distance on C only under the condition that l(C)=|V(LTQ_n)|. In parallel and distributed computing, if cycles of different lengths can be embedded, we can adjust the number of simulated processors and increase the flexibility of demand. This paper demonstrates that in LTQ_n, the cycle embedding containing any two distinct vertices with a feasible distance is extremely flexible.