Oriented discrepancy of Hamilton cycles and paths in digraphs

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 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.