Characterization of \mathcal{R}(2K_2,F_n) with Minimum Order for Small n - Fajri, Muhammad Rafif; Assiyatun, Hilda; Baskoro, Edy Tri (Journal of the Indonesian Mathematical Society, 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 $\mathcal{R}(G, H)$ denote the set of all graphs $F$ satisfying $F → (G, H)$ and for every $e \in E(F),$ $(F - e) \not→ (G, H).$ In this paper, we propose some properties for a graph $G$ of minimum order that belongs to $\mathcal{R}(2K_2,F_n),$ for $n ≥ 3$. We have also found all members of $\mathcal{R}(2K_2,F_n)$ with a minimum order for $n \in [3,7]$.