The order of hypotraceable oriented graphs

A digraph of order nn is hypotraceable   if it is nontraceable but all its induced subdigraphs of order n−1n−1 are traceable. Grötschel et al. (1980) [M. Grötschel, C. Thomassen, Y. Wakabayashi, Hypotraceable digraphs, J. Graph Theory 4 (1980) 377–381] constructed an infinite family of hypotraceable oriented graphs, the smallest of which has order 13. We show that there exist hypotraceable oriented graphs of order nn for every n≥8n≥8 except possibly for n=9,11n=9,11 and that K2¯ is the only one of order less than 8.Furthermore, we determine all the hypotraceable oriented graphs of order 8 and explain the relevance of these results to the problem of determining, for given k≥2k≥2, the maximum order of nontraceable oriented digraphs each of whose induced subdigraphs of order kk is traceable.