An Augmented Pancyclicity Problem of Crossed Cubes.

A graph G is pancyclic if it contains a cycle C of every length with 3 <= l(C) <= |V(G)|, where l(C) denotes the length of C and V(G) denotes the number of vertices in G. In this paper, we propose an augmented pancyclicity problem for the n-dimensional crossed cube CQ(n), which is a popular variant of the hypercube network. Let dC (u, v) denote the distance between any two distinct vertices u and v traversed by a cycle C in CQn, n >= 4. Then, for any integer m with [N +1/2] +1 <= m <= 2(n-1), there exist cycles C of various lengths in CQ(n) such that d(c) (u, v) = m C, where (i) 2m + 1 <= l (C) <= 2(n) if n is odd and , inverted left perpendicular n + 1 /2 inverted right perpendicular +1 2 and (ii) 2m <= l(C) <= 2(n) otherwise.This result indicates that any two distinct vertices of crossed cubes can be embedded on cycles of various feasible lengths with keeping any feasible distance from each other.