Remarks on the distribution of colors in Gallai colorings

Abstract

A Gallai coloring of a complete graph Kn is an edge coloring without triangles colored with three different colors. A sequence e1≥⋯≥ek of positive integers is an (n,k)-sequence if ∑i=1kei=n2. An (n,k)-sequence is a G-sequence if there is a Gallai coloring of Kn with k colors such that there are ei edges of color i for all i,1≤i≤k. Gyárfás, Pálvölgyi, Patkós and Wales proved that for any integer k≥3 there exists an integer g(k) such that every (n,k)-sequence is a G-sequence if and only if n≥g(k). They showed that g(3)=5,g(4)=8 and 2k−2≤g(k)≤8k2+1.We show that g(5)=10 and give almost matching lower and upper bounds for g(k) by showing that with suitable constants α,β>0, αk1.5lnk≤g(k)≤βk1.5 for all sufficiently large k.