Matching-Star Ramsey Minimal Graphs

Abstract

Let G and H be any graphs without isolated vertices. The Ramsey set \({\mathcal{R}(G,H)}\) consists of all graphs F without isolated vertices such that \({F \rightarrow (G,H)}\) and \({F-e \nrightarrow (G,H)}\) for every \({e \in E(F)}\). In this paper, we give necessary and sufficient conditions of graphs belonging to the set \({\mathcal{R}(3K_2,K_{1,n})}\), for any n ≥ 3. Furthermore, by using computational approach, we determine all Ramsey \({(3K_2,K_{1,n})}\)-minimal graphs of order at most 10 vertices for \({3 \leq n \leq 7}\). We construct two classes of bipartite graphs in \({\mathcal{R}(3K_2,K_{1,n})}\), for any \({n \geq 3}\). We also present a class of graphs containing a clique of six vertices in this set. We give large classes of graphs in the set \({\mathcal{R}(t K_2, K_{1,n})}\), for \({n \geq 3}\) and \({t \geq 4}\), that are constructed from t disjoint stars.