Oriented discrepancy of Hamilton cycles

TL;DR

This paper extends Dirac's theorem by conjecturing and approximately proving that high minimum degree in a graph ensures the existence of Hamilton cycles with many edges oriented uniformly, also analyzing random graphs.

Contribution

It introduces a new conjecture on oriented Hamilton cycles, proves an approximate version, and studies the problem in random graphs near the Hamiltonicity threshold.

Findings

Approximate version of the conjecture proven.

High minimum degree guarantees many edges in the same orientation.

Random graphs above threshold likely contain Hamilton cycles with predominantly one orientation.

Abstract

We propose the following conjecture extending Dirac's theorem: if $G$ is a graph with $n\ge 3$ vertices and minimum degree $\delta(G)\ge n/2$, then in every orientation of $G$ there is a Hamilton cycle with at least $\delta(G)$ edges oriented in the same direction. We prove an approximate version of this conjecture, showing that minimum degree $n/2 + O(k)$ guarantees a Hamilton cycle with at least $(n+k)/2$ edges oriented in the same direction. We also study the analogous problem for random graphs, showing that if the edge probability $p = p(n)$ is above the Hamiltonicity threshold, then, with high probability, in every orientation of $G \sim G(n,p)$ there is a Hamilton cycle with $(1-o(1))n$ edges oriented in the same direction.