Oriented Discrepancy of The Square of Hamilton Cycles

TL;DR

This paper investigates the existence of Hamilton cycle squares with large orientation imbalance in large oriented graphs under minimum degree conditions, extending previous Dirac-type results.

Contribution

It establishes that for large graphs with minimum degree at least two-thirds of the vertices, the square of a Hamilton cycle with significant orientation discrepancy exists.

Findings

Proves existence of Hamilton cycle squares with large discrepancy in graphs with minimum degree ≥ 2n/3.

Extends Dirac-type Hamilton cycle results to oriented graphs and their powers.

Provides bounds on the maximum discrepancy based on degree and size.

Abstract

For an oriented graph $G$, the oriented discrepancy problem concerns the existence of a spanning subgraph of $G$ with a large imbalance between its forward and backward edge orientations. Freschi and Lo proved the Dirac-type Hamilton cycle result in oriented graphs, and asked for an analogue for powers of Hamilton cycles under a minimum-degree condition. We show that, for sufficiently large $n$, every oriented graph $G$ on $n$ vertices with minimum degree $\delta(G)\geq 2n/3$ contains the square of a Hamilton cycle $H$ with $\sigma_{\max}(H)$ guaranteed to exceed a function depending on $\delta(G)$ and $n$.