Complete path embeddings in crossed cubes

Crossed cubes are popular variants of hypercubes. In this paper, we study path embeddings between any two distinct nodes in crossed cubes. We prove two important results in the n-dimensional crossed cube: (a) for any two nodes, all paths whose lengths are greater than or equal to the distance between the two nodes plus 2 can be embedded between the two nodes with dilation 1; (b) for any two integers n ⩾ 2 and l   with 1⩽l⩽n+12-1, there always exist two nodes x and y whose distance is l, such that no path of length l + 1 can be embedded between x and y with dilation 1. The obtained results are optimal in the sense that the dilations of path embeddings are all 1. The results are also complete, because the embeddings of paths of all possible lengths between any two nodes are considered.