On k-anti-traceability of oriented graphs

An oriented path P is called anti-directed if every two consecutive arcs of P have opposite orientations. An oriented graph is called k-anti-traceable if every subdigraph induced by k vertices has a hamiltonian anti-directed path. We introduce and study a conjecture, which claims that for every integer $k\ge 2$ there is a least integer $f\left(k\right)$ such that each k-anti-traceable oriented graph on $f\left(k\right)$ vertices has a hamiltonian anti-directed path. We determine $f\left(2\right),f\left(3\right),f\left(4\right)$ and show that every k-anti-traceable oriented graph on sufficiently large number n of vertices admits an anti-directed path that contains all but $o\left(n\right)$ vertices.