On Ramsey (Cn, H)-minimal graphs

Let F, G and H be given simple graphs. The notation F → (G, H) means that for any red-blue coloring of all edges on F there is either a red copy of G or a blue copy of H as a subgraph of F. A graph F is called as a Ramsey (G, H)-minimal if it satisfies the following two conditions: (i) F → (G, H), and (ii) F − e ↛ (G, H) for any e ∈ F . The set of all Ramsey (G, H)-minimal graphs is denoted by R(G, H). In this paper, we study on the graphs in R(G, H) if G is a cycle Cn and H is a path P 3. In particular, we construct some classes of such Ramsey minimal graphs.