Gallai 3-colourings of random graphs
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866918429767761920 |
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| author | Benevides, Fabrício S. Monteiro, Rubens C. S. Mota, Guilherme O. |
| author_facet | Benevides, Fabrício S. Monteiro, Rubens C. S. Mota, Guilherme O. |
| contents | 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 $p \geq Cn^{-1/2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_04115 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Gallai 3-colourings of random graphs Benevides, Fabrício S. Monteiro, Rubens C. S. Mota, Guilherme O. Combinatorics 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 $p \geq Cn^{-1/2}$. |
| title | Gallai 3-colourings of random graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2604.04115 |