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

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• 作者：Meng Liu ; ^{mathliumeng@163.com" class="auth_mail" title="E-mail the corresponding author} ; Yusheng Li ^{li_yusheng@tongji.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Ramsey goodness ; Odd cycle ; Fan and book ; Stability Lemma ; Regularity Lemma
• 刊名：Discrete Mathematics
• 出版年：2016
• 出版时间：6 October 2016
• 年：2016
• 卷：339
• 期：10
• 页码：2481-2489
• 全文大小：383 K

文摘

For graphs G$G$ and H$H$, the Ramsey number R(G,H)$R(G,H)$ is the smallest positive integer N$N$ such that any red/blue edge coloring of K_N$K_N$ contains either a red G$G$ or a blue H$H$. Let G+H$G+H$ be the graph obtained from disjoint G$G$ and H$H$ by adding edges connecting G$G$ and H$H$ completely. It is shown that R(C_2m+1,K_p+nK₁)=2(n+p−1)+1$R(C_{2m+1},K_p+n{K}_1)=2(n+p-1)+1$ and R(C_2m+1,K₁+nH)=2hn+1$R(C_{2m+1},K_1+nH)=2hn+1$, where m,p≥1$m,p\ge 1$ and H$H$ of order h$h$ are fixed and n$n$ is large. Our tools for proofs are Regularity Lemma and Stability Lemma.