Semi-degree threshold for anti-directed Hamiltonian cycles

In 1980, Grant initiated the study of 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 n 2 + 1 , then D contains an anti-directed Hamiltonian cycle. This result is sharp.