On Ramsey (mK2, H)-Minimal Graphs

Abstract

Let R(G, H) denote the set of all graphs F satisfying F → (G, H) and for every e ∈ E(F), (F − e) (G, H). In this paper, we derive the necessary and sufficient conditions for graphs belonging to R(mK2, H) for any graph H and each positive integer m. We give all disconnected graphs in R(mK2, H), for any connected graph H. Furthermore, we prove that if F ∈ R(mK2, P3), then any graph obtained by subdividing one non-pendant edge in F will be in R((m + 1)K2, P3)