Optimal Embeddings of Paths with Various Lengths in Twisted Cubes

Abstract

Twisted cubes are variants of hypercubes. In this paper, we study the optimal embeddings of paths of all possible lengths between two arbitrary distinct nodes in twisted cubes. We use TQ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> to denote the n-dimensional twisted cube and use dist(TQ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , u, v) to denote the distance between two nodes u and v in TQ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , where n ges l is an odd integer. The original contributions of this paper are as follows: 1) We prove that a path of length l can be embedded between u and v with dilation 1 for any two distinct nodes u and v and any integer l with dist(TQ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , u, v) + 2 les l les 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> - 1 (n ges 3) and 2) we find that there exist two nodes u and v such that no path of length dist(TQ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , u, v) + l can be embedded between u and v with dilation 1 (n ges 3). The special cases for the nonexistence and existence of embeddings of paths between nodes u and v and with length dist(TQ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , u, v) + 1 are also discussed. The embeddings discussed in this paper are optimal in the sense that they have dilation 1