Oriented Discrepancy of The Square of Hamilton Cycles

For an oriented graph $G$, the oriented discrepancy problem concerns the existence of a spanning subgraph of $G$ with a large imbalance between its forward and backward edge orientations. Freschi and Lo proved the Dirac-type Hamilton cycle result in oriented graphs, and asked for an analogue for powers of Hamilton cycles under a minimum-degree condition. We show that, for sufficiently large $n$, every oriented graph $G$ on $n$ vertices with minimum degree $\delta (G)\ge 2n\mathrm{/}3$ contains the square of a Hamilton cycle $H$ with ${\sigma }_{\max }(H)$ guaranteed to exceed a function depending on $\delta (G)$ and $n$.