Gallai-Ramsey multiplicity for rainbow small trees

Abstract

Let $G,H$ be two non-empty graphs and $k$ be a positive integer. The Gallai-Ramsey number ${\mathrm{gr}}_k(G:H)$ 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 ${\mathrm{GM}}_k(G:H)$ is defined as the minimum total number of rainbow subgraphs $G$ and monochromatic subgraphs $H$ for all $k$-edge-colored $K_{{\mathrm{gr}}_k(G:H)}$. 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.