Gallai-Ramsey Multiplicity for Rainbow Small Trees

Let G , H be two non-empty graphs and k be a positive integer. The Gallai-Ramsey number gr k ( G : H ) is defined as the minimum positive integer N such that for all n ≥ N , every k -edge-coloring of K n contains either a rainbow subgraph G or a monochromatic subgraph H . The Gallai-Ramsey multiplicity GM k ( G : H ) is defined as the minimum total number of rainbow subgraphs G and monochromatic subgraphs H for all k -edge-colored K gr k ( G : H ) . In this paper, we get some exact values of the Gallai-Ramsey multiplicity for rainbow small trees versus general monochromatic graphs under a sufficiently large number of colors. We also study the bipartite Gallai-Ramsey multiplicity.