On Ramsey (2K₂, 2P_n)-minimal graphs

Let G and H be two given graphs. The notation F → (G,H) means that any red-blue coloring on the edges of F will create either a red subgraph G or a blue subgraph H in F. A graph F is a Ramsey (G,H)-minimal graph if F satisfies two conditions: (1) F → (G,H), and (2) F* ↛ (G,H) for any proper subgraph F* ⊂ F. Denote by R(G,H) the set of all (G,H)-minimal graphs. In this paper, we give necessary conditions for the members of R(2K₂, 2P_n) for n ≥ 3. We prove that 3P_n and F ∪ G are the only disconnected graphs in R(2K₂, 2P_n) for any connected graphs F,G ϵ R(2K₂,P_n). In particular, we determine all graphs in R(2K₂, 2P₃).