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

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