# Ramsey and Gallai-Ramsey numbers for stars with extra independent edges

## TL;DR

This paper investigates Gallai-Ramsey numbers for a class of graphs derived from stars with added independent edges, providing bounds and exact results especially for small cases.

## Contribution

It establishes general bounds and sharp results for Gallai-Ramsey numbers of star-based graphs with extra edges, extending previous understanding.

## Key findings

- Derived bounds for Gallai-Ramsey numbers of $S_t^{r}$ graphs.
- Proved sharp results for the case when $t=2$.
- Extended the theory of Gallai-Ramsey numbers to new graph classes.

## Abstract

Given a graph $G$ and a positive integer $k$, define the \emph{Gallai-Ramsey number} to be the minimum number of vertices $n$ such that any $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) triangle or a monochromatic copy of $G$. In this paper, we obtain general upper and lower bounds on the Gallai-Ramsey numbers for the graph $G = S_t^{r}$ obtained from a star of order $t$ by adding $r$ extra independent edges between leaves of the star so there are $r$ triangles and $t - 2r - 1$ pendent edges in $S_t^{r}$. We also prove some sharp results when $t = 2$.