Ramsey numbers of long cycles versus books or wheels

Given two graphs G 1 and G 2 , denote by G 1 ∗ G 2 the graph obtained from G 1 ∪ G 2 by joining all the vertices of G 1 to the vertices of G 2 . The Ramsey number R ( G 1 , G 2 ) is the smallest positive integer n such that every graph G of order n contains a copy of G 1 or its complement G c contains a copy of G 2 . It is shown that the Ramsey number of a book B m = K 2 ∗ K m c versus a cycle C n of order n satisfies R ( B m , C n ) = 2 n − 1 for n > ( 6 m + 7 ) / 4 which improves a result of Faudree et al., and the Ramsey number of a cycle C n versus a wheel W m = K 1 ∗ C m satisfies R ( C n , W m ) = 2 n − 1 for even m and n ≥ 3 m / 2 + 1 and R ( C n , W m ) = 3 n − 2 for odd m > 1 and n ≥ 3 m / 2 + 1 or n > max { m + 1 , 70 } or n ≥ max { m , 83 } which improves a result of Surahmat et al. and also confirms their conjecture for large n . As consequences, Ramsey numbers of other sparse graphs are also obtained.