The Ramsey numbers of wheels versus odd cycles

Given two graphs G 1 and G 2 , the Ramsey number R ( G 1 , G 2 ) is the smallest integer N such that for any graph G of order N , either G contains G 1 or its complement contains G 2 . Let C m denote a cycle of order m and W n a wheel of order n + 1 . In this paper, it is shown that R ( W n , C m ) = 2 n + 1 for m odd, n ≥ 3 ( m − 1 ) / 2 and ( m , n ) ≠ ( 3 , 3 ) , ( 3 , 4 ) , and R ( W n , C m ) = 3 m − 2 for m , n odd and m < n ≤ 3 ( m − 1 ) / 2 .