Outpaths in semicomplete multipartite digraphs (Q1302162)

An outpath in a digraph is a directed path in which the first vertex \(x\) of the path dominates the last vertex \(z\) of the path only if \(z\) also dominates \(x\). The author shows that if \(x\) is any vertex of a strongly connected semicomplete \(n\)-partite digraph where \(n\geq 3\), then there is an outpath of length \(k\) starting at \(x\) for each \(k\) such that \(2\leq k\leq n-1\). He also shows that if \(xy\) is any arc of a regular \(n\)-partite tournament where \(n\geq 3\), then there is an outpath of length \(k\) starting at \(xy\) for each \(k\) such that \(2\leq k\leq n-1\).