Embedding of meshes in Möbius cubes

Abstract

The hypercube is one of the most popular interconnection networks since it has simple structure and is easy to implement. Möbius cubes form a class of hypercube variants that give better performance with the same number of edges and vertices. In this paper, we consider embedding of meshes in Möbius cubes. The main results obtained in this paper are: (1) For n ≥ 1 , there exists a 2 × 2 n − 1 mesh that can be embedded in the n -dimensional Möbius cube with dilation 1 and expansion 1. (2) For n ≥ 4 , there exists a 4 × 2 n − 2 mesh that can be embedded in the n -dimensional Möbius cube with dilation 2 and expansion 1. (3) For n ≥ 4 , there are two disjoint 4 × 2 n − 3 meshes that can be embedded in the 0-type n -dimensional Möbius cube with dilation 1. (4) For n ≥ 4 , there are two disjoint 4 × 2 n − 3 meshes that can be embedded in the 1-type n -dimensional Möbius cube with dilation 2. Results of (1) and (3) are optimal in the sense that the dilations of the embeddings are 1.