Star-critical Ramsey Numbers of Wheels Versus Odd Cycles

Let K-1,K- (k) be a star of order k + 1 and K-n coproduct K-1,K- (k) the graph obtained from a complete graph K-n and an additional vertex v by joining v to k vertices of K-n. For graphs G and H, the star-critical Ramsey number r(*)(G, H) is the minimum integer k such that any red/blue edge-coloring of Kr-1 coproduct K-1,K- k contains a red copy of G or a blue copy of H, where r is the classical Ramsey number R(G, H). Let C-m denote a cycle of order m and W-n a wheel of order n + 1. Hook (2010) proved that r(*) (W-n, C-3) = n + 3 for n >= 6. In this paper, we show that r(*) (W-n, C-m) = n + 3 for m odd, m >= 5 and n >= 3(m - 1)/2 + 2.