A Gallai coloring of a complete graph $K_n$ is an edge coloring without triangles colored with three different colors. A sequence $e_1\ge \dots \ge e_k$ of positive integers is an $(n,k)$-sequence if $\sum_{i=1}^k e_i=\binom{n}{2}$. An $(n,k)$-sequence is a G-sequence if there is a Gallai coloring of $K_n$ with $k$ colors such that there are $e_i$ edges of color $i$ for all $i,1\le i \le k$. Gyarfas, Palvolgyi, Patkos and Wales proved that for any integer $k\ge 3$ there exists an integer $g(k)$ such that every $(n,k)$-sequence is a G-sequence if and only if $n\ge g(k)$. They showed that $g(3)=5, g(4)=8$ and $2k-2\le g(k)\le 8k^2+1$. We show that $g(5)=10$ and give almost matching lower and upper bounds for $g(k)$ by showing that with suitable constants $\alpha,\beta>0$, $\frac{\alpha k^{1.5}}{\ln k}\le g(k) \le \beta k^{1.5}$ for all sufficiently large $k$.

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关于加莱着色中颜色分布的备注

完整图$K_n$ 的Gallai 着色是没有用三种不同颜色着色的三角形的边着色。如果 $\sum_{i=1}^k e_i=\binom{n}{2}$，则正整数序列 $e_1\ge\dots\ge e_k$ 是 $(n,k)$-序列。$(n,k)$-sequence 是 G-sequence，如果 $K_n$ 的加莱着色具有 $k$ 颜色，使得所有 $i,1\ 都有颜色 $i$ 的 $e_i$ 边\ le i \le k$。Gyarfas、Palvolgyi、Patkos 和 Wales 证明了对于任何整数 $k\ge 3$ 都存在一个整数 $g(k)$ 使得每个 $(n,k)$-序列都是 G-序列当且仅当 $ n\ge g(k)$。他们表明 $g(3)=5, g(4)=8$ 和 $2k-2\le g(k)\le 8k^2+1$。我们证明了 $g(5)=10$ 并给出了几乎匹配的 $g(k)$ 的下限和上限，方法是用合适的常数 $\alpha,\beta>0$, $\frac{\alpha k^ {1.5}}{\ln k}\le g(k) \le \beta k^{1.