Remarks on the distribution of colors in Gallai colorings

TL;DR

This paper investigates the distribution of colors in Gallai colorings of complete graphs, establishing bounds on the minimum number of vertices needed for all color distributions to be realizable.

Contribution

The authors determine the exact value of g(5) and provide nearly tight bounds for g(k), extending previous results on color distribution in Gallai colorings.

Findings

g(5)=10

Bounds for g(k): rac{ ext{constant} imes k^{1.5}}{ ext{log} k} ext{ to } ext{constant} imes k^{1.5}

Extended understanding of color distribution thresholds 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 $k$.