Path embedding in star graphs

The star graph interconnection network has been introduced as an attractive alternative to the hypercube network. In this paper, we consider the path embedding problem in star graphs. Assume that n⩾4n⩾4. We prove that paths of all even lengths from d(x,y)d(x,y) to n!-2n!-2 can be embedded between two arbitrary vertices x and y from the same partite set in the n  -dimensional star graph. In addition, paths of all odd lengths from d(x,y)d(x,y) to n!-1n!-1 can be embedded between two arbitrary vertices x and y from different partite sets in the n-dimensional star graph except that if x and y are adjacent, there is no path of length 3 between them. The result is optimal in the sense that paths of all possible lengths are found in star graphs.