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 \({\text {gr}}_k(G:H)\) is defined as the minimum positive integer N such that for all \(n\ge N\), every k-edge-coloring of \(K_n\) contains either a rainbow subgraph G or a monochromatic subgraph H. The Gallai-Ramsey multiplicity \({\text {GM}}_k(G:H)\) is defined as the minimum total number of rainbow subgraphs G and monochromatic subgraphs H for all k-edge-colored \(K_{{\text {gr}}_k(G:H)}\). 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.