Connected size Ramsey number for matchings vs. small stars or cycles

Abstract

The notation $$F\rightarrow (G,H)$$ means that if the edges of F are colored red and blue, then the red subgraph contains a copy of G or the blue subgraph contains a copy of H. The connected size Ramsey number $$\hat{r}_c(G,H)$$ of graphs G and H is the minimum size of a connected graph F satisfying $$F\rightarrow (G,H)$$ . For $$m \ge 2,$$ the graph consisting of m independent edges is called a matching and is denoted by $$mK_2$$ . In 1981, Erdos and Faudree determined the size Ramsey numbers for the pair $$(mK_2, K_{1,t})$$ . They showed that the disconnected graph $$mK_{1,t} \rightarrow (mK_2,K_{1,t})$$ for $$ t,m \ge 1$$ . In this paper, we will determine the connected size Ramsey number $$\hat{r}_c(nK_2, K_{1,3})$$ for $$n\ge 2$$ and $$\hat{r}_c(3K_2, C_4)$$ . We also derive an upper bound of the connected size Ramsey number $$\hat{r}_c(nK_2, C_4),$$ for $$n\ge 4$$ .