Semi-Degree Threshold for Anti-Directed Hamiltonian Cycles

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, then D contains a directed Hamiltonian cycle. For directed graphs one may ask for other orientations of a Hamiltonian cycle and in 1980 Grant initiated the problem of determining minimum degree conditions for a directed graph D to contain an 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 out-degree and in-degree at least n2 + 1, then D contains an anti-directed Hamiltonian cycle. In fact, we prove the stronger result that n2 is sufficient unless D is one of two counterexamples. This result is sharp.