  In 1960, Ghouila-Houri extended Dirac's theorem to directed graphs by proving
that if D is a directed graph on n vertices with minimum out-degree and
in-degree at least n/2 (i.e. minimum semi-degree at least n/2), then D contains
a directed Hamiltonian cycle. Of course there are other orientations of a cycle
in a directed graph and it is not clear that the semi-degree threshold for the
directed Hamiltonian cycle is the same as the semi-degree threshold for some
other orientation. In 1980, Grant initiated the problem of determining the
minimum semi-degree threshold for the anti-directed Hamiltonian cycle (an
orientation in which consecutive edges alternate direction). We prove that for
sufficiently large even n, if D is a directed graph on n vertices with minimum
semi-degree at least n/2+1, then D contains an anti-directed Hamiltonian cycle.
This result is sharp.
