Oriented discrepancy of Hamilton cycles

We propose the following extension of Dirac's theorem: if (Formula presented.) is a graph with (Formula presented.) vertices and minimum degree (Formula presented.), then in every orientation of (Formula presented.) there is a Hamilton cycle with at least (Formula presented.) edges oriented in the same direction. We prove an approximate version of this conjecture, showing that minimum degree (Formula presented.) guarantees a Hamilton cycle with at least (Formula presented.) edges oriented in the same direction. We also study the analogous problem for random graphs, showing that if the edge probability (Formula presented.) is above the Hamiltonicity threshold, then, with high probability, in every orientation of (Formula presented.) there is a Hamilton cycle with (Formula presented.) edges oriented in the same direction.