Unicyclic Ramsey (P 3, Pn )-minimal graphs obtained from trees in the same class

Abstract

If G, H and F are finite and simple graphs, notation F → (G, H) means that for any red-blue coloring of the edges of F, there is either a red subgraph isomorphic to G or a blue subgraph isomorphic to H. A graph F is a Ramsey (G, H)-minimal graph if F → (G, H) and for every e ∈ E(F), graph F − e ↛ (G, H). The class of all Ramsey (G, H)-minimal graphs (up to isomorphism) will be denoted by R(G, H). The characterization of all graphs in the infinite class R(P 3, Pn ) is still open, for any n ≥ 4. In this paper, we find an infinite families of trees in R(P 3, P 5). We determine how to construct unicyclic graphs in R(P 3, Pn ), for any n ≥ 5 from trees in the same class. Further, we give some properties for the unicyclic graphs constructed from trees in R(P 3, Pn ), for any n ≥ 5.