Colouring Directed Paths in Symmetric Trees, with an Application to Optical Networks

Let T be a symmetric directed tree, i.e., an undirected tree with each edge viewed as two opposite arcs. We prove that the minimum number of colours needed to colour the set of all directed paths in T so that no two paths of the same colour use the same arc of T is equal to the maximum number of paths passing through an arc of T. The proof implies a polynomial algorithm for actually assigning the minimum number of colours. When only a subset of the directed paths is to be coloured, the problem is known to be NP-complete; we describe certain instances of the problem which can be eeciently solved. These results are applied to WDM (wavelength-division multiplexing) routing in all-optical networks. In particular, we solve the all-to-all gossiping problem in optical networks.