Some Multi-Color Ramsey Numbers on Stars versus Path, Cycle or Wheel

Abstract

For given simple graphs $H_1,H_2,\ldots ,H_t$, the Ramsey number $R(H_1,H_2,\ldots ,H_t)$, which is often called multi-color Ramsey number, is the smallest integer n such that for an arbitrary decomposition $\{G_i\}_{i=1}^t$ of the complete graph $K_n$, there is at least one $G_i$ has a subgraph isomorphic to $H_i$. Let $m,n_1,n_2,\ldots , n_t$ be positive integers and $\Sigma =\sum _{i=1}^t(n_i-1)$. Raeisi and Zaghian obtained the $R(K_{1,n_1},\ldots ,K_{1,n_t},C_m)$ and $R(K_{1,n_1},\ldots ,K_{1,n_t},W_m)$ for odd $m\le \Sigma +2$. In this paper, we establish $R(K_{1,n_1},\ldots ,K_{1,n_t},W_m)$ for odd $m\ge \Sigma +3$ and even $m\ge 2\Sigma +2$. We also determine the rest values of $R(K_{1,n_1},\ldots ,K_{1,n_t},C_m)$ except for even $m\le \Sigma +1$ and $R(K_{1,n_1},\ldots ,K_{1,n_t},P_m)$ for $m\ge \Sigma +1$, or $m\le \Sigma$ and $\Sigma \equiv 0,1(\text{ mod }\, m-1)$, which extends a result on $R(K_{1,n_1},\ldots ,K_{1,n_t},P_m)$ obtained by K. Zhang and S. Zhang.