Counting Gallai 3-colorings of complete graphs

TL;DR

This paper establishes an improved upper bound on the number of Gallai 3-colorings of complete graphs, which are edge colorings avoiding rainbow triangles, advancing understanding in graph coloring combinatorics.

Contribution

It provides a tighter upper bound on Gallai 3-colorings of complete graphs, improving previous estimates and contributing to combinatorial graph theory.

Findings

Upper bound of 7(n+1)*2^{n choose 2} for Gallai 3-colorings

Improved over previous bound of (3/2)*(n-1)!*2^{(n-1) choose 2}

Advances understanding of Gallai colorings in 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^{n choose 2}, which improves the best known upper bound of \frac{3}{2} * (n-1)! * 2^{(n-1) choose 2} in [Discrete Mathematics, 2017].