An oriented discrepancy version of Dirac's theorem

TL;DR

This paper resolves a conjecture about the existence of Hamilton cycles with large forward edge discrepancy in oriented graphs with high minimum degree, extending Dirac's theorem to an oriented setting.

Contribution

It provides a complete proof confirming the conjecture that high minimum degree in oriented graphs guarantees Hamilton cycles with many forward edges.

Findings

Confirmed the conjecture for all oriented graphs with minimum degree at least n/2.

Established conditions for Hamilton cycles with large forward edge discrepancy.

Extended Dirac's theorem to an oriented graph setting.

Abstract

The study of graph discrepancy problems, initiated by Erd\H{o}s in the 1960s, has received renewed attention in recent years. In general, given a $2$-edge-coloured graph $G$, one is interested in embedding a copy of a graph $H$ in $G$ with large discrepancy (i.e. the copy of $H$ contains significantly more than half of its edges in one colour). Motivated by this line of research, Gishboliner, Krivelevich and Michaeli considered an oriented version of graph discrepancy for Hamilton cycles. In particular, they conjectured the following generalization of Dirac's theorem: if $G$ is an oriented graph on $n\geq3$ vertices with $\delta(G)\geq n/2$, then $G$ contains a Hamilton cycle with at least $\delta(G)$ edges pointing forward. In this paper, we present a full resolution to this conjecture.