Colour-biased Hamilton cycles in randomly perturbed graphs

TL;DR

This paper studies the existence of colour-biased Hamilton cycles in randomly perturbed graphs, showing that adding a linear number of random edges typically guarantees such cycles under certain minimum degree conditions.

Contribution

It extends previous results by analyzing how adding random edges to dense graphs influences the existence of colour-biased Hamilton cycles in any edge-colouring.

Findings

Adding O(n) random edges to dense graphs typically yields colour-biased Hamilton cycles.

Reducing random edges below a threshold can eliminate colour-biased Hamilton cycles.

At a critical density, adding random edges results in Hamilton cycles with linear colour bias.

Abstract

Given a graph $G$ and an $r$-edge-colouring $\chi$ on $E(G)$, a Hamilton cycle $H\subset G$ is said to have $t$ colour-bias if $H$ contains $n/r+t$ edges of the same colour in $\chi$. Freschi, Hyde, Lada and Treglown showed every $r$-coloured graph $G$ on $n$ vertices with $\delta(G)\geq(r+1)n/2r+t$ contains a Hamilton cycle $H$ with $\Omega(t)$ colour-bias, generalizing a result of Balogh, Csaba, Jing and Pluh\'{a}r. In 2022, Gishboliner, Krivelevich and Michaeli proved that the random graph $G(n,m)$ with $m\geq(1/2+\varepsilon)n\log n$ typically admits an $\Omega(n)$ colour biased Hamilton cycle in any $r$-colouring. In this paper, we investigate colour-biased Hamilton cycles in randomly perturbed graphs. We show that for every $\alpha>0$, adding $m=O(n)$ random edges to a graph $G_\alpha$ with $\delta(G_\alpha)\geq \alpha n$ typically ensures a Hamilton cycle with $\Omega(n)$ colour bias in any $r$-colouring of $G_\alpha\cup G(n,m)$. Conversely, for certain $G_{\alpha}$, reducing the number of random edges to $m=o(n)$ may eliminate all colour biased Hamilton cycles of $G(n,m)\cup G$ in a certain colouring. In contrast, at the critical endpoint $\alpha=(r+1)/2r$, adding $m$ random edges typically results in a Hamilton cycle with $\Omega(m)$ colour-bias for any $1\ll m\leq n$.