Remarks on the distribution of colors in Gallai colorings

Abstract

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 $i,1\le i \le k$. Gy\'arf\'as, P\'alv\"olgyi, Patk\'os 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 $k$.