Gallai 3-colourings of random graphs

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 Erdős-Rényi random graph, by proving that for every $δ> 0$ there are $c$ and $C$ such that with high probability the number of Gallai 3-colourings of $G(n,p)$ is at least $3^{(1-δ)\binom{n}{2}p}$ for $p \leq cn^{-1/2}$, and at most $2^{(1+δ)\binom{n}{2}p}$ for $p \geq Cn^{-1/2}$.

Theorems & Definitions (15)

• Theorem 1
• Lemma 2
• Lemma 3: Embedding lemma for subgraphs of random graphs conlon2014klr
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 7: DeMarco_2011, Theorem 1
• proof : Proof of Lemma \ref{['lemma:main1']}
• Lemma 8: benevides2017edge, Lemma 4.2
• proof : Proof of Lemma \ref{['lemma:main2']}