Two infinite classes of unicyclic Ramsey (P3, P4) – Minimal graphs

Let G and H be simple graphs, the notation F → (G, H) means that for any red-blue coloring of all the edges of graph F implies F contains either a red copy of G or a blue copy of H. A graph F is Ramsey (G, H)-minimal graph if F satisfies two conditions i.e. F is Ramsey (G, H)-graph and F has minimality property. Graph F is called the Ramsey (G, H)-graph if F → (G, H). Meanwhile, F is said to have a minimality property if for any edge e in F then F – e → (G, H). The set of all Ramsey minimal graphs for pair (G, H) is denoted by R(G, H). The Ramsey set for pair (G, H) can be classified in Ramsey-finite or Ramsey-infinite depending on whether R(G, H) is finite or infinite, respectively. Several articles have discussed the problem of determining whether R(G, H) is finite (infinite). It is known that the set R(Pm, Pn), for 3 ≤ m ≤ n is Ramsey-infinite. Some partial results in R(P3, P4) have been obtained. In this paper, we determine two infinite classes of unicyclic graphs in R(P3, P4).