Distribution of colors in Gallai colorings

Abstract

A Gallai coloring is an edge coloring that avoids triangles colored with three different colors. Given integers e1≥e2≥⋯≥ek with ∑i=1kei=n2 for some n, does there exist a Gallai k-coloring of Kn with ei 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 e1−ek≤1 and k≤⌊n∕2⌋. We prove that for any integer k≥3 there is a (unique) integer g(k) with the following property: there exists a Gallai k-coloring of Kn with ei edges in color i for every e1≤⋯≤ek satisfying ∑i=1kei=n2, if and only if n≥g(k). We show that g(3)=5, g(4)=8, and 2k−2≤g(k)≤8k2+1 for every k≥3.