Oriented discrepancy of Hamilton cycles

We propose the following extension of Dirac's theorem: if G $G$ is a graph with n ≥ 3 $n\ge 3$ vertices and minimum degree δ ( ) ∕ 2 $\delta (G)\ge n\unicode{x02215}2$ , then in every orientation there Hamilton cycle at least (G)$ edges oriented same direction. prove an approximate version this conjecture, showing that + 8 k $\frac{n+8k}{2}$ guarantees $(n+k)\unicode{x02215}2$ also study analogous problem for random graphs, edge probability p = $p=p(n)$ above Hamiltonicity threshold, then, high probability, ~ $G\unicode{x0007E}G(n,p)$ 1 − o $(1-o(1))n$