Ramsey $(K_{1,2},C_4)$-Minimal Graphs

Abstract

For any given graphs $G$ and $H$, we write $F\to (G,H)$ to mean that any red-blue coloring of the edges of $F$ contains a red copy of $G$ or a blue copy of $H$. A graph $F$ is $(G,H)$-minimal (Ramsey-minimal) if $F\to (G,H)$ but $F^{\ast }\nrightarrow (G,H)$ for any proper subgraph $F^{\ast }\subset F$. The class of all $(G,H)$-minimal graphs is denoted by $\mathcal{R}(G,H)$. In this paper, we will determine the graphs in $\mathcal{R}(K_{1,2},C_4)$.