An Ore-type Theorem for Oriented Discrepancy of Hamilton Cycles

Abstract: Oriented graph discrepancy problems focus on finding specific subgraphs within a given oriented graph $G$ that contain a significant number of edges in one direction. This concept was first introduced by Gishboliner, Krivelevich, and Michaeli, and has since been further investigated by Freschi and Lo [J. Combin. Theory, Ser. B 169 (2024)], who gave a tight lower bound for the discrepancy of Hamilton cycles in terms of the minimum degree of $G$. Furthermore, they raised the problem of extending such results to Ore-type conditions. Here, an Ore-type condition refers to the minimum degree-sum of non-adjacent vertices, formally defined as: $$σ_2(G)=\min{d(x)+d(y)\mid x, y \in V(G) \text{ and } xy \notin E(G)}$$. In this paper, we address this question by showing that for every sufficiently large oriented graph $G$, if $σ_2(G)\geq n$, then $G$ contains a Hamilton cycle $C$ with at least $\max{n/2,σ_2(G)/2-o(n)}$ edges in one direction. Moreover, this result is asymptotically tight.