Embedding of Cycles in Twisted Cubes with Edge-Pancyclic

Abstract

In this paper, we study the embedding of cycles in twisted cubes. It has been proven in the literature that, for any integer l, 4≤l≤2 n , a cycle of length l can be embedded with dilation 1 in an n-dimensional twisted cube, n≥3. We obtain a stronger result of embedding of cycles with edge-pancyclic. We prove that, for any integer l, 4≤l≤2 n , and a given edge (x,y) in an n-dimensional twisted cube, n≥3, a cycle C of length l can be embedded with dilation 1 in the n-dimensional twisted cube such that (x,y) is in C in the twisted cube. Based on the proof of the edge-pancyclicity of twisted cubes, we further provide an O(llog l+n 2+nl) algorithm to find a cycle C of length l that contains (u,v) in TQ n for any (u,v)∈E(TQ n ) and any integer l with 4≤l≤2 n .