Gallai 3-colourings of random graphs

TL;DR

This paper investigates the number of Gallai 3-colourings in Erdős-Rényi random graphs, establishing bounds that depend on the edge probability and providing initial estimates for such colourings.

Contribution

It provides the first bounds on the number of Gallai 3-colourings in random graphs, relating the count to the edge probability and high probability events.

Findings

Number of Gallai 3-colourings is at least exponential in the number of edges for small p.

Number of Gallai 3-colourings is at most exponential in the number of edges for larger p.

High probability bounds are established for the counts based on p and constants c, C.

Abstract

A Gallai $k$-colouring of a graph $G$ is a colouring of $E(G)$ with $k$ colours that induces no rainbow triangles, that is, a triangle with edges of 3 different colours. We give a first step towards estimating the number of Gallai colourings of the Erd\H{o}s-R\'enyi random graph, by proving that for every $\delta > 0$ there are $c$ and $C$ such that with high probability the number of Gallai 3-colourings of $G(n,p)$ is at least $3^{(1-\delta)\binom{n}{2}p}$ for $p \leq cn^{-1/2}$, and at most $2^{(1+\delta)\binom{n}{2}p}$ for $p \geq Cn^{-1/2}$.