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 \(m \geq 2n-1 \geq 7\). With the help of computers, we obtain the exact values of \(14\) small cycle-book Ramsey numbers.