Ramsey numbers of long cycles versus books or wheels

Given two graphs G1G1 and G2G2, denote by G1∗G2G1∗G2 the graph obtained from G1∪G2G1∪G2 by joining all the vertices of G1G1 to the vertices of G2G2. The Ramsey number R(G1,G2)R(G1,G2) is the smallest positive integer nn such that every graph GG of order nn contains a copy of G1G1 or its complement GcGc contains a copy of G2G2. It is shown that the Ramsey number of a book Bm=K2∗Kmc versus a cycle CnCn of order nn satisfies R(Bm,Cn)=2n−1R(Bm,Cn)=2n−1 for n>(6m+7)/4n>(6m+7)/4 which improves a result of Faudree et al., and the Ramsey number of a cycle CnCn versus a wheel Wm=K1∗CmWm=K1∗Cm satisfies R(Cn,Wm)=2n−1R(Cn,Wm)=2n−1 for even mm and n≥3m/2+1n≥3m/2+1 and R(Cn,Wm)=3n−2R(Cn,Wm)=3n−2 for odd m>1m>1 andn≥3m/2+1n≥3m/2+1 or n>max{m+1,70}n>max{m+1,70} or n≥max{m,83}n≥max{m,83} which improves a result of Surahmat et al. and also confirms their conjecture for large nn. As consequences, Ramsey numbers of other sparse graphs are also obtained.