Counting Gallai 3-colorings of complete graphs

Abstract

An edge coloring of the $n$-vertex complete graph $K_n$ is a Gallai coloring if it does not contain any rainbow triangle, that is, a triangle whose edges are colored with three distinct colors. We prove that the number of Gallai colorings of $K_n$ with at most three colors is at most $7(n+1)2^{\left(\genfrac{}{}{0ex}{}{n}{2}\right)}$, which improves the best known upper bound of $\frac{3}{2}(n-1)!\cdot 2^{\left(\genfrac{}{}{0ex}{}{n-1}{2}\right)}$ in Benevides et al. (2017) .