Multirate Rearrangeable Clos Networks and a Generalized Edge Coloring Problem on Bipartite Graphs

... that the minimum number m(n, r) of middle-state switches for the symmetric 3-stage Clos network C(n, m(n, r), r) to be rearrangeable in the multirate enviroment is at most 2n − 1. This problem is equivalent to a generalized version of the bipartite graph edge coloring problem. The best bounds known so far on the function m(n, r) is 11n/9 ≤ m(n, r) ≤ 41n/16 + O(1), for n, r ≥ 2, derived by Du-Gao-Hwang-Kim (SIAM J. Comput., 28, 1999). In this paper, we make several contributions. Firstly, we give evidence to show that even a stronger result might hold. In particular, we give a coloring algorithm to show that m(n, r) ≤ ⌈(r + 1)n/2⌉, which implies m(n, 2) ≤ ⌈3n/2⌉- stronger than the conjectured value of 2n − 1. Secondly, we derive that m(2, r) = 3 by an elegant argument. Lastly, we improve both the best upper and lower bounds given above: ⌈5n/4 ⌉ ≤ m(n, r) ≤ 2n − 1 + ⌈(r − 1)/2⌉, where the upper bound is an improvement over 41n/16 when r is relatively small compared to n. We also conjecture that m(n, r) ≤ ⌊2n(1 − 1/2 r)⌋