Traceability of kk-traceable oriented graphs

A digraph of order at least kk is kk-traceable if each of its subdigraphs of order kk is traceable. We note that 2-traceable oriented graphs are tournaments and for k≥3k≥3, kk-traceable oriented graphs can be regarded as generalized tournaments. We show that for 2≤k≤62≤k≤6 every kk-traceable oriented graph is traceable, thus extending the well-known fact that every tournament is traceable. This result does not extend to k=7k=7. In fact, for every k≥7k≥7, except possibly for k=8k=8 or 10, there exist kk-traceable oriented graphs that are nontraceable. However, we show that for every k≥2k≥2 there exists a smallest integer t(k)t(k) such that every kk-traceable oriented graph of order at least t(k)t(k) is traceable.