Ramsey numbers of a fixed odd-cycle and generalized books and fans

Abstract

For graphs G and H, the Ramsey number R(G,H) is the smallest positive integer N such that any red/blue edge coloring of KN contains either a red G or a blue H. Let G+H be the graph obtained from disjoint G and H by adding edges connecting G and H completely. It is shown that R(C2m+1,Kp+nK1)=2(n+p−1)+1 and R(C2m+1,K1+nH)=2hn+1, where m,p≥1 and H of order h are fixed and n is large. Our tools for proofs are Regularity Lemma and Stability Lemma.