On Ramsey (4K2,P3)-minimal graphs

Let F,G, and H be simple graphs. We write F→(G,H) to mean that any red-blue coloring of all edges of F will contain either a red copy of G or a blue copy of H. A graph F (without isolated vertices) satisfying F→(G,H) and for each e∈E(F),(F−e)↛(G,H) is called a Ramsey (G,H)-minimal graph. The set of all Ramsey (G,H)-minimal graphs is denoted by ℛ(G,H). In this paper, we derive the necessary and sufficient condition of graphs belonging to ℛ(4K2,H), for any connected graph H. Moreover, we give a relation between Ramsey (4K2,P3)- and (3K2,P3)-minimal graphs, and Ramsey (4K2,P3)- and (2K2,P3)-minimal graphs. Furthermore, we determine all graphs in ℛ(4K2,P3).