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

TL;DR

This paper explores the oriented discrepancy of Hamilton cycles in oriented graphs under Ore-type conditions, extending previous Dirac-type results and proposing conjectures supported by new proofs.

Contribution

It introduces two conjectures extending Hamilton cycle discrepancy results to Ore-type conditions and provides partial proofs supporting these conjectures.

Findings

Proposed conjectures for Hamilton cycle discrepancy in oriented graphs under Ore conditions

Partial proofs supporting the conjectures

Extension of Dirac's theorem to Ore-type conditions in oriented graphs

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 $\delta (D) \ge n/ 2$, then $D$ contains a Hamilton oriented cycle with at least $\delta(D)$ 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.