Colouring Paths in Directed Symmetric

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. This result is applied to solve the all-to-all communication problem in wavelength{ division{multiplexing (WDM) routing in all{optical networks, that is, we give an eecient algorithm to optimally assign wavelengths to the all the paths of a tree network. It is known that the problem of colouring a general subset of all possible paths in a symmetric directed tree is an NP-hard problem. We study conditions for a given set S of paths be coloured eeciently with the minimum possible number of colours/wavelengths.