Erd{\H o}s (1963) initiated extensive graph discrepancy research on 2-edge-colored graphs. Gishboliner, Krivelevich, and Michaeli (2023) launched similar research on oriented graphs. They conjectured the following generalization of Dirac's theorem: If the minimum degree $\delta$ of an $n$-vertex oriented graph $G$ is greater or equal to $n/2$,then $G$ has a Hamilton oriented cycle with at least $\delta$ forward arcs. This conjecture was proved by Freschi and Lo (2024) who posed an open problem to extend their result to an Ore-type condition. We propose two conjectures for such extensions and prove some results which provide support to the conjectures. For forward arc maximization on Hamilton oriented cycles and paths in semicomplete multipartite digraphs and locally semicomplete digraphs, we obtain characterizations which lead to polynomial-time algorithms.

翻译：Erdős (1963) 开创了关于二边着色图的广泛图偏差研究。Gishboliner、Krivelevich 和 Michaeli (2023) 在有向图上开启了类似研究。他们提出了以下狄拉克定理的推广猜想：若一个 $n$ 阶有向图 $G$ 的最小度 $\delta$ 大于等于 $n/2$，则 $G$ 存在一条至少包含 $\delta$ 条前向弧的哈密顿有向圈。Freschi 和 Lo (2024) 证明了该猜想，并提出了一个开放问题：将其结果推广至 Ore 型条件。我们为此类推广提出了两个猜想，并证明了一些支持这些猜想的结果。针对半完全多部有向图与局部半完全有向图中哈密顿有向圈与路径的前向弧最大化问题，我们得到了可导出多项式时间算法的刻画性结论。