Abstract
Let D be a digraph. A path partition of D is called k-optimal if the sum of the k-norms of its paths is minimal. The k-norm of a path P is min(|V(P)|,k). Berge’s path partition conjecture claims that for every k-optimal path partition P there are k disjoint stable sets orthogonal to P. For general digraphs the conjecture has been proven for k=1,2,λ−1,λ, where λ is the length of a longest path in the digraph. In this paper we prove the conjecture for λ−2 and λ−3.
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