# Distribution of colors in Gallai colorings

## TL;DR

This paper investigates the distribution of edge colors in Gallai colorings of complete graphs, establishing conditions for their existence and identifying a threshold size for such colorings with given color distributions.

## Contribution

It provides new sufficient and necessary conditions for the existence of Gallai colorings with specified edge distributions, including thresholds based on the number of colors.

## Key findings

- Existence of Gallai-colorings when the largest and smallest color edges differ by at most 1 and k ≤ ⌊n/2⌋.
- Existence of a threshold function g(k) determining when all distributions are realizable.
- Exact values of g(k) for small k and bounds for larger k.

## Abstract

A Gallai coloring is an edge coloring that avoids triangles colored with three different colors. Given integers $e_1\ge e_2 \ge \dots \ge e_k$ with $\sum_{i=1}^ke_i={n \choose 2}$ for some $n$, does there exist a Gallai $k$-coloring of $K_n$ with $e_i$ edges in color $i$? In this paper, we give several sufficient conditions and one necessary condition to guarantee a positive answer to the above question. In particular, we prove the existence of a Gallai-coloring if $e_1-e_k\le 1$ and $k \le \lfloor n/2\rfloor$. We prove that for any integer $k\ge 3$ there is a (unique) integer $g(k)$ with the following property: there exists a Gallai $k$-coloring of $K_n$ with $e_i$ edges in color $i$ for every $e_1\le\dots \le e_k$ satisfying $\sum_{i=1}^ke_i={n\choose 2}$, if and only if $n\ge g(k)$. We show that $g(3)=5$, $g(4)=8$, and $2k-2\le g(k)\le 8k^2+1$ for every $k\ge 3$.