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.