On Ore-type conditions for oriented discrepancy of Hamilton cycles

The oriented discrepancy of Hamiltonian cycles in a Hamiltonian graph $ G $ of order $ n $ is the largest integer $ \ell $ such that every orientation $ D $ of $ G $ contains a Hamiltonian oriented cycle $ C $ satisfying $ \sigma_{\max}(C)\ge\lceil(n+\ell)/2\rceil $, where $ \sigma_{\max}(C) $ denotes the maximum of the numbers of forward and backward arcs of $ C $. For a vertex $ w $ in an oriented graph $ D $, let $ d(w) $ denote the degree of $ w $ in the underlying graph of $ D $. Ai, Guo, Gutin, Lan, Shao, Yeo and Zhou [AIMS Math. 11 (2026), 24106-24119] proved that, for any positive integer $ k $, every oriented graph $ D $ of order $ n\ge30+4(k-1) $ with $ s^{*}(D)\ge8k $ contains a Hamilton oriented cycle $ C $ satisfying $ \sigma_{\max}(C)\ge\lceil(n+k)/2\rceil $, where $ s^{*}(D) $ is the minimum of $ d(u)+d(v)-n $ over all distinct non-adjacent vertices $ u, v $ of $ D $ when $ D $ is not a tournament, while $ s^{*}(D) = n-2 $ when $ D $ is a tournament. They asked whether the factor $ 8 $ in the bound $ s^{*}(D)\ge8k $ can be reduced. For every $ k\ge6 $, we prove that the aforementioned conclusion follows from $ s^{*}(D)\ge4k-3 $; however, when $ n+k $ is even and $ k\ge7 $, then the condition $ s^{*}(D)\ge4k-7 $ is sufficient. We also obtain results for $ 1\le k\le5 $. Moreover, for every fixed positive integer $ k $ and all sufficiently large $ n $, we prove that the same conclusion follows from $ s^{*}(D)\ge3k $ when $ n+k $ is odd and from $ s^{*}(D)\ge3k-3 $ when $ n+k $ is even.