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Let G be an acyclic directed graph with \V(G)\>k. We prove that there exists a colouring { C x , C 2 ,..., C m } such that for every collection {P l9 P 2 ,... ,P k } of k vertex disjoint paths with |UjLi Pj\ a maximum, each colour class C, meets min{|CJ, k} of these paths. An analogous theorem, partially interchanging the roles of paths and colour classes, has been shown by Cameron [4] and Saks [17] and we indicate a third proof. Introduction.. ,ι>/) will be denoted by K(P). The cardinality of P, denoted by |P|, is |F(P)|.A family 0* of paths is called a path-partition of G if its members are vertex disjoint and U{F(P): Pe#) == V. For each nonnegative integer k 9 the k-norm \0>\ k of a path partition ^ = {P v ... y P m } is defined by A partition which minimizes 1^)^. is called k-optimum. For example, a 1-optimum partition is a partition P containing a minimum number of paths.A partial k-colouring is a family #* = {C^, C 2 ,... ,C r } of at most /: disjoint independent sets C f called colour classes. The cardinality of a partial ^-colouring #* = {C 1? C 2 ,...,C t ) is |U{ βl Q, and ... ,P m } and a partial fc-colouring ^A are orthogonal if every path P. in 0> meets min{ (P^, fc} different colour classes of %> k .Berge [2] made the following conjecture:Conjecture 1. Let G be a directed graph and let A: be a positive integer. Then for every fc-optimum path partition ^, there exists a partial A>colouring #* orthogonal to 0*.

Analysis of GPS traces shows that people often do not use the least cost path through the transportation network while making trips. This leads to the question which structural path characteristics can be used to construct realistic route choice sets for use in traffic simulation models. In this paper, we investigate the hypothesis that, for utilitarian trips, the route between origin and destination consists of a small number of concatenated least cost paths. The hypothesis is verified by analyzing routes extracted from large sets of recorded GPS traces which constitute revealed preference information. Trips have been extracted from the traces and for each trip the path in the transportation network is determined by map matching. This is followed by a path decomposition phase for which the algorithm constitutes the first contribution of this paper. There are multiple ways to split a given path in a directed graph into a minimal number of subpaths of minimal cost

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