Matching-Star Ramsey Minimal Graphs

Let G and H be any graphs without isolated vertices. The Ramsey set $${mathcal{R}(G,H)}$$R(G,H) consists of all graphs F without isolated vertices such that $${F rightarrow (G,H)}$$F→(G,H) and $${F-e nrightarrow(G,H)}$$F-e↛(G,H) for every $${e in E(F)}$$e∈E(F). In this paper, we give necessary and sufficient conditions of graphs belonging to the set $${mathcal{R}(3K_2,K_{1,n})}$$R(3K2,K1,n), for any n ≥ 3. Furthermore, by using computational approach, we determine all Ramsey $${(3K_2,K_{1,n})}$$(3K2,K1,n)-minimal graphs of order at most 10 vertices for $${3 leq n leq 7}$$3≤n≤7. We construct two classes of bipartite graphs in $${mathcal{R}(3K_2,K_{1,n})}$$R(3K2,K1n), for any $${n geq 3}$$n≥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 $${mathcal{R}(t K_2, K_{1,n})}$$R(tK2,K1,n), for $${n geq 3}$$n≥3 and $${t geq 4}$$t≥4, that are constructed from t disjoint stars. © 2015, Springer Basel.