Let $G_1$ be the acyclic tournament with the topological sort $0\lt 1 \lt 2 \lt \dots \lt n \lt n+1$ defined on node set $N\cup \{0,n+1\}$, where $N=\{1,2,\dots,n\}$. The arcs of $G_1$ are therefore the pairs $(i,j)$, with $i \lt j$. Consider the (di)graph $G_2$ obtained by taking two copies of every arc in $G_1$ and colouring one blue and the other red. A 2-colour partition of $N$ (2-CP, for short) is a pair of a blue and a red (di)paths, both from $0$ to $n+1$, such that each node of $N$ is included in exactly one path. If each blue and red arc has a real cost, the cost of a 2-CP is the sum of the costs of the arcs in the two paths. We show that a minimum cost 2-CP may be found in polynomial time, and give a complete description of the convex hull of the incidence vectors of 2-colour partitions of $N$. There is a natural generalization of 2-CPs when $k\gt 2$ colours are present. We state some results on $k$-CPs.

Key words: Acyclic tournaments, shortest paths, integer polytopes.