On the Ramsey Numbers $R(C_m,B_n)$

Abstract

For given graphs $G_1$ and $G_2$, the Ramsey number $R(G_1,G_2)$ is defined to be the least positive integer $n$ such that every graph $G$ on $n$ vertices, either $G$ contains a copy of $G_1$ or the complement of $G$ contains a copy of $G_2$. In this note, we show that $R(C_m,B_n)=2m-1$ for $m\ge 2n-1\ge 7$. With the help of computers, we obtain the exact values of $14$ small cycle-book Ramsey numbers.