On Star-critical (K1,n,K1,m + e) 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 $K_n\to (G,H)$. The Ramsey number $r(G,H)$ is defined as the smallest positive integer $n$ such that $K_n\to (G,H)$. Star-critical Ramsey number $r_{\ast }(G,H)$ is defined as the largest value of $k$ such that $K_{r(G,H)-1}\bigsqcup K_{1,k}\to (G,H)$. In this paper, we will find $r_{\ast }(K_{1,n},K_{1,m}+e)$ for all $n,m\ge 3$.

8 pages, 5 figures