An edge coloring of the n-vertex complete graph, $K_n$, is a Gallai coloring if it does not contain any rainbow triangle, that is, a triangle whose edges are colored with three distinct colors. We prove that for n large and every k with $k\le 2^{n/4300}$, the number of Gallai colorings of $K_n$ that use at most k given colors is $(\left(\begin{array}{c}k\\ 2\end{array}\right)+o_n(1))2^{\left(\begin{array}{c}n\\ 2\end{array}\right)}$. Our result is asymptotically best possible and implies that, for those k, almost all Gallai k-colorings use only two colors. However, this is not true for $k\ge 2^{n/2}$.

中文翻译：

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完全图的Gallai k着色数

n-顶点完成图的边缘着色，${ķ}_{ñ}$如果不包含任何彩虹三角形（即，其边缘被三种不同颜色着色的三角形），则为Gallai着色。我们证明了对ñ大，每ķ用$ķ\le 2^{ñ/4300}$，加莱着色的数量 ${ķ}_{ñ}$最多使用k个给定的颜色是$（（\begin{array}{c}ķ\\ 2\end{array}）+{Ø}_{ñ}（\mathrm{1个}））2^{（\begin{array}{c}ñ\\ 2\end{array}）}$。我们的结果是渐近最佳的，并暗示对于那些k，几乎所有Gallai k色仅使用两种颜色。但是，这不适用于$ķ\ge 2^{ñ/2}$。