Characterization of R(2K_2,F_n) with Minimum Order for Small n - Fajri, Muhammad Rafif; Assiyatun, Hilda; Baskoro, Edy Tri (IndoMS, 2025)
A Fan graph F_n is defined as the graph P_n+K_1, where P_n is the path on n vertices. The notation F → (G, H) means that if all edges of F are arbitrarily colored by red or blue, then either the subgraph of F induced by all red edges contains a graph G or the subgraph of F induced by all blue edges contains a graph H. Let R(G, H) denote the set of all graphs F satisfying F → (G, H) and for every e E(F), (F - e) → (G, H). In this paper, we propose some properties for a graph G of minimum order that belongs to R(2K_2,F_n), for n ≥ 3. We have also found all members of R(2K_2,F_n) with a minimum order for n [3,7].