On k-Optimum Dipath Partitions and Partial k-Colourings of Acyclic Digraphs

For an acyclic digraph G and positive integer k, we give a min-max equality for: $minimum\{k<\wp <+<V(G)-\cup \wp \}:\wp isanode-disjointsetofdipathsinG\}$ We prove that there exist k independent sets {S_j : j = 1, 2,..., k} such that for every optimum ℘, every P∈℘; intersects each S_j, and every dipath P in G−∪℘; intersects |P| different sets S_j.