Colour-biased Hamilton cycles in dense graphs and random graphs

A classical result of Dirac says that every $n$-vertex graph with minimum degree at least $\frac{n}{2}$ contains a Hamilton cycle. A `discrepancy' version of Dirac's theorem was shown by Balogh--Csaba--Jing--Pluh\'ar, Freschi--Hyde--Lada--Treglown, and Gishboliner--Krivelevich--Michaeli as follows. Every $r$-colouring of the edge set of every $n$-vertex graph with minimum degree at least $(\frac{1}{2} + \frac{1}{2r} + o(1))n$ contains a Hamilton cycle where one of the colours appears at least $(1+o(1))\frac{n}{r}$ times. In this paper, we generalize this result by asymptotically determining the maximum possible value $f_{r,\alpha}(n)$ for every $\alpha \in [\frac{1}{2}, 1]$ such that every $r$-colouring of the edge set of every $n$-vertex graph with minimum degree at least $\alpha n$ contains a Hamilton cycle where one of the colours appears at least $f_{r,\alpha}(n)$ times. In particular, we show that $f_{r,\alpha}(n) = (1-o(1)) \min\{(2\alpha - 1)n, \frac{2\alpha n}{r}, \frac{2n}{r+1}\}$ for every $\alpha\in [\frac{1}{2} + \frac{1}{2r}, 1]$. A graph $H$ is called an $\alpha$-residual subgraph of a graph $G$ if $d_H(v)\ge \alpha d_{G[V(H)]}(v)$ for every $v\in V(H)$. Extending Dirac's theorem in the setting of random graphs, Lee and Sudakov showed the following. The Erd\H{o}s--R\'enyi random graph $G(n,p)$, with $p$ above the Hamiltonicity threshold, typically has the property that every $(\frac{1}{2} +o(1))$-residual spanning subgraph contains a Hamilton cycle. Motivated by this, we prove the following random version of our `discrepancy' result. The random graph $G \sim G(n,p)$, with $p$ above the Hamiltonicity threshold, typically satisfies that every $r$-colouring of the edge set of every $\alpha$-residual spanning subgraph of $G$ contains a Hamilton cycle where one of the colours appears at least $f_{r,\alpha}(n)$ times.