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 extension of Dirac's theorem: If $D$ is an oriented graph on $n \ge 3$ vertices with minimum degree $δ(D) \ge n/ 2$, then $D$ contains a Hamilton oriented cycle with at least $δ(D)$ arcs in the same direction. 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 results which provide support to the conjectures.

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) 在有向图上展开了类似研究。他们提出了狄拉克定理的如下推广猜想：若 $D$ 是顶点数 $n \ge 3$ 的有向图，且最小度 $δ(D) \ge n/ 2$，则 $D$ 包含一个哈密顿有向环，其中至少有 $δ(D)$ 条弧朝向相同方向。Freschi 和 Lo (2024) 证明了这个猜想，并提出一个开放性问题：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，并证明了支持这些猜想的结果。