Abstract
Let G be an acyclic directed graph with \V(G)\>k. We prove that there exists a colouring { Cx, C2,..., Cm } such that for every collection {Pl9 P2,... ,Pk} 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.
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