Optimal stability results on color-biased Hamilton cycles

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.