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

Chung and Ross (SIAM J. Comput., 20, 1991) conjectured 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 this 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 show that m(n; r) d(r+1)n=2e, which impliesm(n; 2) d3n=2e 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: d5n=4e m(n; r) 2n 1 + d(r 1)=2e, 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) b2n(1 1=2r) .