Constrained Bipartite Edge Coloring with Applications to Wavelength Routing

Motivated by the problem of efficient 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 αL different colors, then the rest of the edges can be colored using at most (1+α/2)L colors. We also show that this bound is tight by constructing instances in which (1+α/2)L colors are indeed necessary. We also obtain tight bounds on the number of colors that each pair of opposite vertices can see.