On Ramsey (3K₂,P₃)-minimal graphs

For any given two graphs G and H, we use the notation F→(G,H) to mean that in any red-blue coloring of the edges of F, the following must hold: F contains either a red subgraph G or a blue subgraph H. A graph F is a Ramsey (G,H)-minimal graph if F→(G,H) but F*↛(G,H) for any proper subgraph F* of F. Let R(G,H) be the class of all Ramsey (G,H)-minimal graphs. In this paper, we derive the properties of graphs belonging to R(3K₂,P₃). By using these properties we determine all graphs belonging to this class.