An Ore-type Theorem for Oriented Discrepancy of Hamilton Cycles

TL;DR

This paper extends discrepancy results for Hamilton cycles in oriented graphs to Ore-type degree conditions, establishing that large graphs with high degree-sum conditions contain Hamilton cycles with many edges in one direction.

Contribution

It proves an Ore-type theorem for oriented graphs ensuring Hamilton cycles with high directional discrepancy, generalizing previous minimum degree results.

Findings

For large oriented graphs with $\sigma_2(G) extgreater= n$, a Hamilton cycle exists with at least $ ext{max}igrace n/2,rac{\sigma_2(G)}{2}-o(n)igrace$ edges in one direction.

The result is asymptotically tight, confirming the optimality of the bound.

Addresses an open problem on extending discrepancy bounds to Ore-type conditions.

Abstract

Oriented graph discrepancy problems focus on finding specific subgraphs within a given oriented graph $G$ that contain a significant number of edges in one direction. This concept was first introduced by Gishboliner, Krivelevich, and Michaeli, and has since been further investigated by Freschi and Lo [J. Combin. Theory, Ser. B 169 (2024)], who gave a tight lower bound for the discrepancy of Hamilton cycles in terms of the minimum degree of $G$. Furthermore, they raised the problem of extending such results to Ore-type conditions. Here, an Ore-type condition refers to the minimum degree-sum of non-adjacent vertices, formally defined as: $\sigma_2(G)=\min\{d(x)+d(y)\mid x, y \in V(G) \text{ and } xy \notin E(G)\}$. In this paper, we address this question by showing that for every sufficiently large oriented graph $G$, if $\sigma_2(G)\geq n$, then $G$ contains a Hamilton cycle $C$ with at least $\max\{n/2,\sigma_2(G)/2-o(n)\}$ edges in one direction. Moreover, this result is asymptotically tight.