# Gallai-Ramsey number of an 8-cycle

## TL;DR

This paper determines the exact Gallai-Ramsey numbers for 8-cycles in k-edge-colored complete graphs, extending previous results that focused mainly on triangles.

## Contribution

It provides the first known precise Gallai-Ramsey numbers for an 8-cycle for all values of k, expanding the understanding of these numbers beyond triangles.

## Key findings

- Exact Gallai-Ramsey numbers for C8 for all k
- Extension of known results from triangles to 8-cycles
- Advancement in combinatorial graph coloring theory

## Abstract

Given graphs $G$ and $H$ and a positive integer $k$, the Gallai-Ramsey number $gr_{k}(G : H)$ is the minimum integer $N$ such that for any integer $n \geq N$, every $k$-edge-coloring of $K_{n}$ contains either a rainbow copy of $G$ or a monochromatic copy of $H$. These numbers have recently been studied for the case when $G = K_{3}$, where still only a few precise numbers are known for all $k$. In this paper, we extend the known precise Gallai-Ramsey numbers to include $H = C_{8}$ for all $k$.