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 $δ&gt; 0$ there are $c$ and $C$ such that with high probability the number of Gallai 3-colourings of $G(n,p)$ is at least $3<sup>{(1-δ)\binom{n}{2}p}$ for $p \leq cn<sup>{-1/2}$, and at most $2<sup>{(1+δ)\binom{n}{2}p}$ for $p \geq Cn<sup>{-1/2}$.

• The paper identifies a sharp threshold at p = Θ(n⁻¹ᐟ²) that distinguishes flexible 3-colouring regimes from rigid 2-colouring behavior.
• It employs probabilistic triangle analysis and the sparse Regularity Lemma to derive asymptotically tight bounds on the number of Gallai 3-colourings.
• The results imply that in sparse regimes, nearly all 3-colourings are valid, while in denser graphs, rainbow-avoidance forces a collapse towards 2-colourings.

Gallai 3-Colourings in Random Graphs

Introduction and Background

This paper undertakes a quantitative study of Gallai $k$-colourings in the context of Erdős–Rényi random graphs, focusing specifically on the case $k = 3$ (2604.04115). A Gallai $k$-colouring of a graph $G$ is an edge-colouring with $k$ colours such that no triangle in $G$ is rainbow (i.e., every triangle has at least two edges of the same colour). Gallai colourings are pivotal in extremal and Ramsey graph theory, connecting to concepts such as Gallai partitions, as well as to information-theoretic and anti-Ramsey results.

Whereas prior work established the prevalence and characteristics of Gallai colourings in dense settings (notably, complete graphs), the enumeration and threshold properties in sparse random graphs $G(n,p)$ remained unaddressed. This paper provides the first probabilistic bounds for the number of Gallai 3-colourings of $G(n,p)$ across a natural range of the edge probability $p$.

Main Results

The principal achievement is the identification of a threshold function at $p = \Theta(n^{-1/2})$ for the transition between the structure and count of Gallai 3-colourings. The main theorem asserts the following, for any fixed $k = 3$0:

• For sparsity $k = 3$1 (for constant $k = 3$2), with high probability

$k = 3$3

where $k = 3$4 denotes the number of Gallai 3-colourings and $k = 3$5 the edge count.

• For density $k = 3$6 (for constant $k = 3$7), with high probability

$k = 3$8

This result demonstrates that, except near the threshold $k = 3$9, the number of Gallai 3-colourings of $k$0 is concentrated near the trivial lower bound $k$1 (essentially all 2-colourings, which are always Gallai) or the naive upper bound $k$2 (all colourings).

Lower Range: Sparse Regime

For $k$3, triangles are scarce within $k$4. Since rainbow triangles are forbidden, but most edges do not participate in triangles, a typical colouring can almost freely assign any of the 3 colours to almost all edges. The proof leverages probabilistic estimates for triangle counts in $k$5—including classical results and a precise concentration bound from DeMarco and Kahn—to show that the fraction of "problematic" edges in triangles is negligible in the exponent.

Upper Range: Dense Regime

For $k$6, triangles proliferate. As a result, the triangle constraint becomes stringent: in almost every Gallai 3-colouring, the actual use of the third colour nearly vanishes. The upper bound is established using the sparse multicolour Szemerédi Regularity Lemma to partition the graph and reduce the problem to bounding the number of reduced colourings. The anti-Ramsey constraints are encoded at the reduced graph level, and an embedding lemma for random graphs is used to rule out rainbow triangles at this level. An entropy argument, based on the counting method for rainbow-avoiding colourings, bounds the number of Gallai colourings above by essentially that of 2-colourings.

Methodological Highlights

• Triangle Analysis in Random Graphs: For very sparse $k$7, classical threshold behaviour and variance calculations show w.h.p. very few triangles, allowing most edges to be coloured arbitrarily.
• Sparse Regularity Lemma: Enables decomposition of random graphs into quasi-random pairs for all $k$8. This reduction is crucial for the upper bound.
• Embedding and Gallai Functions: The embedding lemma ensures that constraints on the reduced graph reflect the global rainbow-freeness property. The counting reduction uses an edge-weighting scheme reflecting the Gallai colouring constraint to tightly cap the number of allowed colourings.

Implications and Discussion

This work provides the first delineation of typical Gallai 3-colouring counts in random graphs and identifies a precise threshold at $k$9 separating the sparse and dense regimes. The results imply that, in random graphs, avoidance of rainbow triangles is "easy" in extremely sparse instances (where triangles are rare) but becomes "difficult" once triangles are common, essentially inheriting the rigidity of 2-colourings.

From a theoretical standpoint, this sharp threshold mirrors known phenomena in other anti-Ramsey and forbidden substructure problems, but the colouring context and rainbow-freeness constraint introduce unique analytics and combinatorial challenges (e.g., the necessity of multicolour regularity and anti-Ramsey embedding).

Future Directions

• The authors conjecture analogous behaviour for Gallai $G$0-colourings for all constant $G$1, i.e., the threshold at $G$2 generalizes, and the extremal exponent for the colouring count remains at $G$3 (rather than $G$4) above threshold.
• Potential improvements could sharpen the error terms, especially near the threshold, or address more refined distributional properties (e.g., typical structure of Gallai colourings for $G$5 near threshold).
• The methodology suggests future use of sparse random versions of containers and more powerful entropy-based counting for general forbidden subgraphs and their colour-avoiding extensions.

Conclusion

The paper gives the first asymptotically tight bounds for the number of Gallai 3-colourings in Erdős–Rényi random graphs, identifying a structural phase transition at $G$6. In the sparse regime, w.h.p. almost all 3-colourings are Gallai; in the dense regime, the count collapses to near that of 2-colourings. These results open the way to deeper enumerative and probabilistic analyses of rainbow-freeness and Gallai constraints in random and pseudorandom settings, with extensions to larger $G$7 and refined random graph models presenting a natural avenue for continued study.