Ore-type extension of the oriented discrepancy theorem

Establish that every oriented graph D on n ≥ 3 vertices with minimum degree δ(D) and d(u) + d(v) ≥ n for every pair of non-adjacent vertices u and v contains a Hamilton oriented cycle C satisfying o_max(C) ≥ max{δ(D), n − δ(D)}.

Background

The paper discusses an Ore-type analogue of the oriented discrepancy theorem of Freschi and Lo, which guarantees a Hamilton oriented cycle with sufficiently many forward arcs under a minimum-degree condition. Conjecture 1.3 replaces the minimum-degree threshold with an Ore-type degree-sum condition while retaining a bound that interpolates between the minimum degree and its complementary quantity n − δ(D).

The authors explicitly state that they were unable to prove the conjecture, although they establish supporting results, including a strengthened degree-sum theorem and constructions showing that the proposed bound is sharp.