Oriented discrepancy of Hamilton cycles in oriented graphs satisfying Ore-type condition

Abstract

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 $\hat{I}´(D)\ge n/2$, then $D$ contains a Hamilton oriented cycle with at least $\hat{I}´(D)$ 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.

We have decided to partition arxiv v. 1 paper into two papers. This is the first one of them