Multi-hop all-to-all optical routings in Cartesian product networks

This paper considers the wavelength assignment problem for Cartesian product networks with multi-hops. An upper bound of the (uniform) wavelength index for Cartesian product networks with single-hop is established. This result leads to a consequence for the nth power of an arbitrary network with k-hops. In particular, if k=1, this bound partially generalizes the results of Pankaj [R.K. Pankaj, Architectures for linear lightwave networks, PhD thesis, Dept. of Electrical Engineering and Computer Science, MIT, Cambridge, MA, 1992], Bermond et al. [J.-C. Bermond, L. Gargano, S. Perennes, A.A. Rescigno, U. Vaccaro, Efficient collective communication in optical networks, Theoret. Comput. Sci. 233 (2000) 165–189] and Beauquier [B. Beauquier, All-to-all communication for some wavelength-routed all-optical networks, Networks 33 (1999) 179–187] for hypercubes and Hamming graphs. As an application, a tight upper bound for Hamming graph with k hops is established and a corresponding open problem is also proposed.