Constrained Bipartite Edge Coloring with Applications to Wavelength Routing

Motivated by the problem of eecient routing in all-optical networks, we study a constrained version of the bipartite edge coloring problem. We show that if the edges adjacent to a pair of opposite vertices of an L-regular bipartite graph are already colored with L diierent colors, then the rest of the edges can be colored using at most (1+=2)L colors. We also show that this bound is tight by constructing instances in which (1 + =2)L colors are indeed necessary. We also obtain tight bounds on the number of colors that each pair of opposite vertices can see. Using the above results, we obtain a polynomial time greedy algorithm that assigns proper wavelengths to a set of requests of maximum load L per directed ber link on a directed ber tree using at most 5=3L wavelengths. This improves previous results of 9, 7, 6, 10]. We also obtain that no greedy algorithm can in general use less than 5=3L wavelengths for a set of requests of load L in a directed ber tree, and thus that our algorithm is optimal in the class of greedy algorithms which includes the algorithms presented in 9, 7, 6, 10].