Bounded monochromatic components for random graphs

TL;DR

This paper investigates the structure of vertex partitions in random graphs, focusing on the size of monochromatic components and introducing parameters like the $t$-component chromatic number and stability number, revealing phase transitions and precise asymptotics.

Contribution

It introduces and analyzes the $t$-component chromatic number and stability number in random graphs, providing thresholds, phase transitions, and exact asymptotic behaviors.

Findings

Threshold for $t$ around $ heta(p^{-1} ext{log} np)$ for phase transition.

Precise asymptotics for $t$-component parameters when $p$ is fixed.

Identification of a non-smooth function characterizing behavior at $t = heta( ext{log} n)$.

Abstract

We consider vertex partitions of the binomial random graph $G_{n,p}$. For $np\to\infty$, we observe the following phenomenon: in any partition into asymptotically fewer than $\chi(G_{n,p})$ parts, i.e. $o(np/\log np)$ parts, one part must induce a connected component of order at least roughly the average part size. Stated another way, we consider the $t$-component chromatic number, the smallest number of colours needed in a colouring of the vertices for which no monochromatic component has more than $t$ vertices. As long as $np \to \infty$, there is a threshold for $t$ around $\Theta(p^{-1}\log np)$: if $t$ is smaller then the $t$-component chromatic number is nearly as large as the chromatic number, while if $t$ is greater then it is around $n/t$. For $0 < p <1$ fixed, we obtain more precise information. We find something more subtle happens at the threshold $t = \Theta(\log n)$, and we determine that the asymptotic first-order behaviour is characterised by a non-smooth function. Moreover, we consider the $t$-component stability number, the maximum order of a vertex subset that induces a subgraph with maximum component order at most $t$, and show that it is concentrated in a constant length interval about an explicitly given formula, so long as $t = O(\log \log n)$. We also consider a related Ramsey-type parameter and use bounds on the component stability number of $G_{n,1/2}$ to describe its basic asymptotic growth.