Embedding Paths of Different Lengths into Crossed Cubes

Crossed cubes are attractive alternatives to the popular hypercubes with many advantageous properties. In this paper, we study embeddings of paths of different lengths between any two distinct nodes in crossed cubes.We prove two important results: (a) Paths of all possible lengths greater than or equal to the distance between any two nodes plus 2 can be embedded between the two nodes with dilation 1; And (b) in the n-dimensional crossed cube, for any two integers n ge 3 and l with 1 le l le leftlceil {frac{{n + 1}}{2}} rightrceil -1, there always exist two nodes x and y, such that the distance between x and y is l and any path of length l + 1 cannot be embedded between x and y with dilation 1. The results improve those provided by Fan, Lin, and Jia.