Size Ramsey numbers of small graphs versus fans or paths

Abstract

For two graphs $G_1$ and $G_2$, the size Ramsey number $\hat{r}(G_1,G_2)$ is the smallest positive integer $m$ for which there exists a graph $G$ of size $m$ such that for any red-blue edge-coloring of the graph $G$, $G$ contains either a red subgraph isomorphic to $G_1$, or a blue subgraph isomorphic to $G_2$. Let $P_n$ be a path with $n$ vertices, $n{K}_2$ a matching with $n$ edges, and $F_n$ a graph with $n$ triangles sharing exactly one vertex. If $G_1$ is a small fixed graph and $G_2$ denotes any graph from a graph class, one can sometimes completely determine $\hat{r}(G_1,G_2)$. Faudree and Sheehan confirmed all size Ramsey numbers of $P_3$ versus complete graphs in 1983. The next year Erdős and Faudree confirmed that of $2K_2$ versus complete graphs and complete bipartite graphs. We obtain three more Ramsey results of this type. For $n\ge 3$, we prove that $\hat{r}(P_3,F_n)=4n+4$ if $n$ is odd, and $\hat{r}(P_3,F_n)=4n+5$ if $n$ is even. This result refutes a conjecture proposed by Baskoro et al. We also show that $\hat{r}(2K_2,F_2)=12$ and $\hat{r}(2K_2,F_n)=5n+3$ for $n\ge 3$. In addition, we prove that $\hat{r}(2K_2,n{P}_m)=\min \{nm+1,(n+1)(m-1)\}$. This result verifies a conjecture posed by Vito and Silaban.