Counting Gallai 3-colorings of complete graphs

An edge coloring of the n-vertex complete graph Kn 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 Kn with at most three colors is at most 7(n+1)2n2, which improves the best known upper bound of 32(n−1)!⋅2n−12 in Benevides et al. (2017) .