Traceability of k-traceable oriented graphs

A digraph of order at least k is k-traceable if each of its subdigraphs of order k is traceable. We note that 2-traceable oriented graphs are tournaments and for k ≥ 3, k-traceable oriented graphs can be regarded as generalized tournaments. We show that for 2 ≤ k ≤ 6 every k-traceable oriented graph is traceable, thus extending the well known fact that every tournament is traceable. This result does not extend to k = 7. In fact, for every k ≥ 7, except possibly for k = 8 or 10, there exist k-traceable oriented graphs that are nontraceable. However, we show that for every k ≥ 2 there exists a smallest integer t(k) such that every k-traceable oriented graph of order at least t(k) is traceable.