Embedding meshes into crossed cubes

Crossed cubes are important variants of hypercubes. In this paper, we consider embeddings of meshes in crossed cubes. The major research findings in this paper are: (1) For any integer n ⩾ 1, a 2 × 2n−1 mesh can be embedded in the n-dimensional crossed cube with dilation 1 and expansion 1. (2) For any integer n ⩾ 4, two node-disjoint 4 × 2n−3 meshes can be embedded in the n-dimensional crossed cube with dilation 1 and expansion 2. The obtained results are optimal in the sense that the dilations of the embeddings are 1. The embedding of the 2 × 2n−1 mesh is also optimal in terms of expansion because it has the smallest expansion 1.