Desi Rahmadani, Edy Tri Baskoro, Hilda Assiyatun, Roslan Hasni
Let F, G and H be the graphs. We write F → (G, H) if in any red-blue coloring of the edges of F, there is 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 F-e→(G,H), for every e ∈ E(F). The set of all Ramsey (G, H) - minimal graphs (up to isomorphism) will be denoted by R(G, H). A pair (G, H) is Ramsey-finite if the set of R(G, H) is finite. Otherwise, the pair (G, H) is Ramsey- infinite. Some partial results for R(P3Pn), for odd n ≥ 7 have been obtained. However, since the set R(P3, Pn), for n ≥ 7 is Ramsey-infinite then it is interesting to know some infinite families in this set. In this paper, we construct some infinite classes of unicyclic graphs in R(P3, Pn), for each odd n ≥ 7. These unicyclic graphs are formed from the odd cycles by connecting every vertex of the cycle to a center vertex of K1,2. This process will be done recursively. © 2024 Author(s).
Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Negeri Malang, Malang, Indonesia; Department of Mathematics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Bandung, Indonesia; Special Interest Group of Modeling and Data Analytics, Faculty of Ocean Engineering Technology and Informatics, Universiti Malaysia Terengganu, Kuala Nerus, Terengganu, 21030, Malaysia