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

Abstract

Let K_n 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 K_n, there is a red copy of graph G or a blue copy of graph H in K_n. Determining the Ramsey number R(C_n, tW_m) 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 C_n is tW_m–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(C_n, 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.