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An Ore-type Theorem for Oriented Discrepancy of Hamilton Cycles

Yufei Chang, Yangyang Cheng, Zhilan Wang, Shuo Wei, Jin Yan

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: $σ_2(G)=\min\{d(x)+d(y)\mid x, y \in V(G) \text{ and } xy \notin E(G)\}$. In this paper, we address this question by showing that for every sufficiently large oriented graph $G$, if $σ_2(G)\geq n$, then $G$ contains a Hamilton cycle $C$ with at least $\max\{n/2,σ_2(G)/2-o(n)\}$ edges in one direction. Moreover, this result is asymptotically tight.

An Ore-type Theorem for Oriented Discrepancy of Hamilton Cycles

Abstract

Oriented graph discrepancy problems focus on finding specific subgraphs within a given oriented graph 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 . 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: . In this paper, we address this question by showing that for every sufficiently large oriented graph , if , then contains a Hamilton cycle with at least edges in one direction. Moreover, this result is asymptotically tight.
Paper Structure (12 sections, 17 theorems, 22 equations)

This paper contains 12 sections, 17 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and proof overview
  3. Notation
  4. The extremal example
  5. Proof Strategy
  6. Useful lemmas and tools
  7. The Diregularity Lemma and related results
  8. Hajnal-Szemerédi Theorem and Probabilistic tool
  9. Proof of \ref{['th2']}
  10. Preparation of \ref{['th2']}
  11. Completion of \ref{['th2']}
  12. Concluding remarks

Key Result

Theorem 1.1

Let $0 < c < 1/4$ and $n \in \mathbb{N}$ be sufficiently large. If $G$ is a $2$-edge-coloured graph on $n$ vertices with $\delta(G) \geq (3/4 + c)n$, then $G$ contains a Hamilton cycle with at least $(1/2 + c/64)n$ edges in one colour.

Theorems & Definitions (28)

  • Theorem 1.1: Bal
  • Theorem 1.2: Freschi
  • Theorem 1.4
  • Proposition 2.1
  • proof
  • Lemma 3.1: Degree form of the Diregularity Lemma
  • Lemma 3.2
  • proof
  • Lemma 3.3: Kelly
  • Lemma 3.4: Komlos1 Blow-up Lemma
  • ...and 18 more