Some families of Ramsey (P4, P5)–minimal graphs

Let F, G and H be connected and simple graphs. The notation F→(G, H) means that for any red-blue coloring of all the edges of F contains either a red copy isomorphic to G or a blue copy isomorphic to H. A graph F is a Ramsey (G, H)-minimal graph if F is Ramsey graph and F satisfies minimality property. A graph F is said to be a Ramsey graph if F→(G, H). The minimality property of F state that 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) is said to be Ramsey-finite or Ramsey-infinite if ℛ(G, H) is finite or infinite, respectively. The set ℛ(Pm, Pn), for 3≤m≤n is Ramsey-infinite, it means that the number of graphs in the set is infinite. Some partial results in ℛ(P4, P4) have been obtained. In this paper, we determine some graphs in ℛ(P4, P5) constructed from graphs in ℛ(P4, P4). We obtain graphs without vertices of degree one and graphs with some vertices of degree one in ℛ(P4, P5). Furthermore, we characterize all graphs on six vertices in ℛ(P4, P5).