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.