Oriented discrepancy of Hamilton cycles

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