Characterization of Ramsey (2K ₂, W ² n )-Minimal Graphs of the Smallest Order*

Let F, G , and H be simple graphs. The notation F → ( G, H ) denotes that for every red-blue edge-coloring of F , there exists either a red subgraph isomorphic to G or a blue subgraph isomorphic to H . A graph F is called a Ramsey ( G, H )- minimal if F → ( G, H ), but for every edge e ∈ E ( F ), we have F − e ↛ ( G, H ). The collection of all such graphs is denoted by ℛ( G, H ). A Windmill graph W m n is defined as the graph K 1 + mK n , where K n is complete graph on n vertices. In this paper, we propose some properties for a graph F of minimum order that belongs to ℛ(2 K 2 , W 2 n ), for n ≥ 3. We have also found all members of ℛ(2 K 2 , W 2 n ) with a minimum order for n ∈ [3, 5].