Optimal stability results on color-biased Hamilton cycles

TL;DR

This paper establishes optimal stability conditions for Hamilton cycles in edge-colored graphs with respect to color-bias, showing that near-extremal graphs must resemble known extremal configurations, with precise degree and bias thresholds.

Contribution

It proves the exact stability thresholds for Hamilton cycles with bounded color-bias in edge-colored graphs, extending and sharpening previous results with optimal constants.

Findings

Optimal stability threshold at minimum degree n/2 + 6r^2 m

Structural characterization of near-extremal graphs

Additive error term Θ(m) is tight for r=2

Abstract

We investigate Hamilton cycles in edge-colored graphs with \( r \) colors, focusing on the notion of color-bias (discrepancy), the maximum deviation from uniform color frequencies along a cycle. Foundational work by Balogh, Csaba, Jing, and Pluh\'{a}r, and the later generalization by Freschi, Hyde, Lada, and Treglown, as well as an independent work by Gishboliner, Krivelevich, and Michaeli, established that any \(n\)-vertex graph with minimum degree exceeding \( \frac{(r+1)n}{2r} + \frac{m}{2}\) contains a Hamilton cycle with color-bias at least \(m\), and characterized the extremal graphs with minimum degree \(\frac{(r+1)n}{2r}\) in which all Hamilton cycles are perfectly balanced. We prove the optimal stability results: for any positive integers \(r\ge 2\) and \( m < 2^{-6} r^{2} n,\) if every Hamilton cycle in an \( n \)-vertex graph with minimum degree exceeding \( \frac{n}{2} + 6r^{2}m \) has color-bias less than \( m \), then the graph must closely resemble the extremal constructions of Freschi, Hyde, Lada, and Treglown. The leading term \( \frac{n}{2} \) in the degree condition is optimal, as it is the sharp threshold for guaranteeing Hamiltonicity. Moreover, we show the additive error term \(\Theta(m)\) is also best possible when \(m\) is large and \(r=2\), since weaker condition \(\frac{n}{2}+o(m)\) allow for a counterexample. Notably, the structural stability threshold \( \frac{1}{2} \) lies strictly below the extremal threshold \( \frac{1}{2} + \frac{1}{2r} \) required to force color imbalance. Our proof leverages local configurations to deduce global structure, revealing a rigid combinatorial dichotomy.