Let $B_n^{ (k)}$ be the book graph which consists of $n$ copies of $K_{ k+1}$ all sharing a common $K_k$, and let $C_m$ be a cycle of length $m$. In this paper, we first determine the exact value of $r(B_n^{ (2)}, C_m)$ for $\frac{ 8}{ 9}n+112\le m\le \lceil\frac{ 3n}{ 2}\rceil+1$ and $n \geq 1000$. This answers a question of Faudree, Rousseau and Sheehan (Cycle--book Ramsey numbers, { \it Ars Combin.,} { \bf 31} (1991), 239--248) in a stronger form when $m$ and $n$ are large. Building upon this exact result, we are able to determine the asymptotic value of $r(B_n^{ (k)}, C_n)$ for each $k \geq 3$. Namely, we prove that for each $k \geq 3$, $r(B_n^{ (k)}, C_n)= (k+1+o_k(1))n.$ This extends a result due to Rousseau and Sheehan (A class of Ramsey problems involving trees, { \it J.~London Math.~Soc.,} { \bf 18} (1978), 392--396).
Categories: math.CO