A note on the Ramsey number for cycle with respect to multiple copies of wheels

Let Kn be a complete graph with n vertices. For graphs G and H, the Ramsey number R(G, H) is the smallest positive integer n such that in every red-blue coloring on the edges of Kn, there is a red copy of graph G or a blue copy of graph H in Kn. Determining the Ramsey number R(Cn, tWm) for any integers t ≥ 1, n ≥ 3 and m ≥ 4 in general is a challenging problem, but we conjecture that for any integers t ≥ 1 and m ≥ 4, there exists n0 = f(t, m) such that cycle Cn is tWm–good for any n ≥ n0. In this paper, we provide some evidence for the conjecture in the case of m = 4 that if n ≥ n0 then the Ramsey number R(Cn, tW4)=2n + t − 2 with n0 = 15t2 − 4t + 2 and t ≥ 1. Furthermore, if G is a disjoint union of cycles then the Ramsey number R(G, tW4) is also derived.