Degree-sum discrepancy bound via the Ore-degree parameter

Prove that every oriented graph D on n ≥ 3 vertices with s*(D) ≥ 0 contains a Hamilton oriented cycle C satisfying o_max(C) ≥ ⌈(n + s*(D))/2⌉, where s*(D) is the minimum value of d(u) + d(v) − n over all non-adjacent vertex pairs, with s*(D) = n − 2 for tournaments.

Background

Conjecture 1.4 proposes a second Ore-type generalization of the oriented discrepancy theorem, using the parameter s*(D), which measures the minimum degree-sum surplus over n among non-adjacent vertex pairs. The conjectured lower bound on the maximum number of forward arcs is expressed directly in terms of this parameter.

The paper states that the conjecture remains unresolved and proves only an approximate version. A family of oriented graphs is also constructed to demonstrate that the proposed bound is tight.