Gallai-Ramsey Multiplicity

TL;DR

This paper investigates the minimum total number of rainbow and monochromatic subgraphs in edge-colored complete graphs, providing bounds for the Gallai-Ramsey multiplicity involving small rainbow subgraphs.

Contribution

It introduces bounds for Gallai-Ramsey multiplicity, extending classical Ramsey theory to account for the number of specific rainbow and monochromatic subgraphs.

Findings

Established upper bounds for Gallai-Ramsey multiplicity.

Derived lower bounds for specific small rainbow subgraphs.

Connected multiplicity bounds to classical Ramsey numbers.

Abstract

Given two graphs $G$ and $H$, the \emph{general $k$-colored Gallai-Ramsey number} $\operatorname{gr}_k(G:H)$ is defined to be the minimum integer $m$ such that every $k$-coloring of the complete graph on $m$ vertices contains either a rainbow copy of $G$ or a monochromatic copy of $H$. Interesting problems arise when one asks how many such rainbow copy of $G$ and monochromatic copy of $H$ must occur. The \emph{Gallai-Ramsey multiplicity} $\operatorname{GM}_{k}(G,H)$ is defined as the minimum total number of rainbow copy of $G$ and monochromatic copy of $H$ in any exact $k$-coloring of $K_{\operatorname{gr}_{k}(G,H)}$. In this paper, we give upper and lower bounds for Gallai-Ramsey multiplicity involving some small rainbow subgraphs.