On Ramsey (P3, C6)-minimal graphs

We write notation F → (G, H) for graphs F, G and H to mean that if there is any two-colouring, say red and blue, of all edges of F, then the red subgraph contains a copy of G or the blue subgraph contains a copy of H. The graph F is Ramsey (G, H)-minimal if F → (G, H) but F − e ↛ (G, H) for any e ∈ E(F). The class of all Ramsey (G, H)-minimal graphs will be denoted by ℜ(G, H). In this paper, we prove that there is only one graph that has 6 vertices and 9 edges in ℜ(P3, C6) and we determine some graphs in ℜ(P3, C6).