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 , t W 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 n 0 = f ( t , m ) such that cycle C n is t W m –good for any n ≥ n 0 . In this paper, we provide some evidence for the conjecture in the case of m = 4 that if n ≥ n 0 then the Ramsey number R ( C n , t W 4 )=2 n + t − 2 with n 0 = 15 t 2 − 4 t + 2 and t ≥ 1 . Furthermore, if G is a disjoint union of cycles then the Ramsey number R ( G , t W 4 ) is also derived.