# On Star-critical (K1,n,K1,m + e) Ramsey numbers

## TL;DR

This paper determines the star-critical Ramsey numbers for specific pairs of graphs involving stars and a single edge, extending understanding of edge-coloring Ramsey properties.

## Contribution

The paper explicitly calculates the star-critical Ramsey numbers for the pairs (K_{1,n}, K_{1,m}+e) for all n,m ≥ 3, a novel extension in Ramsey theory.

## Key findings

- Calculated r_*(K_{1,n}, K_{1,m}+e) for all n,m ≥ 3.
- Extended known results to new graph pairs involving stars and edges.
- Provided exact values for these star-critical Ramsey numbers.

## Abstract

Let $G, H$ be finite graphs without loops or multiple edges and $K_n$ denote the complete graph on $n$ vertices. If for every red/blue colouring of edges of the complete graph $K_n$, there exists a red copy of $G$, or a blue copy of $H$, we will say that $K_n\rightarrow (G,H)$. The Ramsey number $r(G, H)$ is defined as the smallest positive integer $n$ such that $K_{n} \rightarrow (G, H)$. Star-critical Ramsey number $r_*(G, H)$ is defined as the largest value of $k$ such that $K_{r(G,H)-1} \sqcup K_{1,k} \rightarrow (G, H)$. In this paper, we will find $r_*(K_{1,n}, K_{1,m}+e)$ for all $n,m \geq 3$.