On the Ramsey Numbers of Certain Graphs of Order Five versus All Connected Graphs of Order Six

Abstract

The Ramsey numbers $r(F,G)$ are investigated, where $F$ is a non-tree graph of order $5$ and minimum degree $1$, and $G$ is a connected graph of order $6$. For all pairs $(F,G)$ where $F\ne K_5–K_{1,3}$, the exact value of $r(F,G)$ is determined. In order to settle $F=K_5–K_{1,3}$, we prove $r(K_5–K_{1,3},G)=r(K_4,G)$.