Gallai-Ramsey multiplicity for rainbow small trees

TL;DR

This paper investigates the minimum total count of rainbow small trees and monochromatic graphs in edge-colored complete graphs, providing exact values and exploring bipartite cases under many colors.

Contribution

It offers new exact values for Gallai-Ramsey multiplicity involving rainbow small trees and monochromatic graphs, including bipartite cases, for large numbers of colors.

Findings

Exact values of Gallai-Ramsey multiplicity for rainbow small trees versus monochromatic graphs.

Results on bipartite Gallai-Ramsey multiplicity.

Analysis applicable under sufficiently large number of colors.

Abstract

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