Abstract. Let [Formula: see text] be the book graph which consists of [Formula: see text] copies of triangles all sharing a common edge. Let [Formula: see text] be a cycle of length [Formula: see text]. In 1978, Rousseau and Sheehan initiated the study of the book–cycle Ramsey number. A lot of effort has been made to determine the value of [Formula: see text] since then. In [ Ars Combin., 31 (1991), pp. 239–248], Faudree, Rousseau, and Sheehan mentioned the following: “we know practically nothing about [Formula: see text] when [Formula: see text] is even and greater than four. Also, the problem of computing [Formula: see text] when [Formula: see text] is odd and [Formula: see text] and [Formula: see text] are nearly equal provides an unanswered test of strength.” Answering the second part of the question above, the second and fifth authors recently obtained the value of [Formula: see text] for [Formula: see text] and [Formula: see text] being large. However, the value of [Formula: see text] is previously unknown for [Formula: see text] and [Formula: see text] being even as well as [Formula: see text] and [Formula: see text] being odd. In this paper, for even [Formula: see text], we manage to determine the value of [Formula: see text] provided that [Formula: see text] is linear with [Formula: see text] and [Formula: see text] is large enough. Thus this makes progress towards the first part of the question above. In addition, for odd [Formula: see text], we are able to obtain the value of [Formula: see text] for [Formula: see text] and [Formula: see text] being large.

In this paper we determine the chromatic number of graphs with two odd cycle lengths. Let $G$ be a graph and $L(G)$ be the set of all odd cycle lengths of $G$. We prove that (1) if $L(G)=\{3,3+2l\}$, where $l\geq 2$, then $\chi(G)=\max\{3,\omega(G)\}$, and (2) if $L(G)=\{k,k+2l\}$, where $k\geq 5$ and $l\geq 1$, then $\chi(G)=3$. These, together with the case $L(G)=\{3,5\}$ solved in [S.-S. Wang, SIAM J. Discrete Math., 22 (2008), pp. 1040--1072] give a complete solution to the general problem addressed in [S.-S. Wang, SIAM J. Discrete Math., 22 (2008), pp. 1040--1072; S.-M. Camacho and I. Schiermeyer, Discrete Math., 309 (2009), pp. 4916--4919; and T. Kaiser, O. Rucký, and R. Škrekovski, SIAM J. Discrete Math., 25 (2011), pp. 1069--1088]. Our results also improve a classical theorem of Gyárfás which asserts that $\chi(G)\le 2|L(G)|+2$ for any graph $G$.