The typical structure of Gallai colorings and their extremal graphs

TL;DR

This paper investigates the structure and extremal properties of Gallai colorings, showing that almost all such colorings of complete graphs use only two colors and identifying the graphs with the maximum number of Gallai colorings.

Contribution

It characterizes the typical structure of Gallai colorings and determines extremal graphs for the maximum number of Gallai colorings among all n-vertex graphs.

Findings

Almost all Gallai r-colorings of K_n use only 2 colors.

K_n maximizes Gallai 3-colorings among n-vertex graphs for large n.

K_{n/2, n/2} maximizes Gallai r-colorings for r ≥ 4.

Abstract

An edge coloring of a graph $G$ is a Gallai coloring if it contains no rainbow triangle. We show that the number of Gallai $r$-colorings of $K_n$ is $\left(\binom{r}{2}+o(1)\right)2^{\binom{n}{2}}$. This result indicates that almost all Gallai $r$-colorings of $K_n$ use only 2 colors. We also study the extremal behavior of Gallai $r$-colorings among all $n$-vertex graphs. We prove that the complete graph $K_n$ admits the largest number of Gallai $3$-colorings among all $n$-vertex graphs when $n$ is sufficiently large, while for $r\geq 4$, it is the complete bipartite graph $K_{\lfloor n/2 \rfloor, \lceil n/2 \rceil}$. Our main approach is based on the hypergraph container method, developed independently by Balogh, Morris, and Samotij as well as by Saxton and Thomason, together with some stability results for containers.