Ramsey unsaturated and saturated graphs

A graph is Ramsey unsaturated if there exists a proper supergraph of the same order with the same Ramsey number, and Ramsey saturated otherwise. We present some conjectures and results concerning both Ramsey saturated and unsaturated graphs. In particular, we show that cycles Cn and paths Pn on n vertices are Ramsey unsaturated for all n ≥ 5. 1 Results and Conjectures Throughout this article, r(G,H) will denote the Ramsey number of a pair of graphs (G,H), i.e., the minimum n such that in any coloring of the edges of Kn with colors red and blue, we either obtain a red subgraph isomorphic to G, or a blue subgraph isomorphic to H. When G = H the notation is reduced to r(G). For years there has been interest in how the Ramsey number of the complete graph grows as its order increases. A favorite question of Erdős (personal communication) was whether one can show that the difference between two consecutive Ramsey numbers is at least quadratic, i.e., whether r(Kn+1)− r(Kn) ≥ cn for some positive constant c. Since Kim [9] has shown that there is a constant c1 such that r(Kn, K3) ≥ c1n log n it is easy to show r(Kn+2) − r(Kn) ≥ c1n log n . Concerning the question of Erdős, the best that is presently known is that the difference of consecutive Ramsey numbers is at least linear [2]. In this paper another growth question is addressed but of a different nature. Given a graph G, is there a nontrivial supergraph of the same order with the same Ramsey number? Also ∗Research partially supported by NSF grant EIA-0130352 †University of Memphis, Department of Mathematics, Dunn Hall, 3725 Norriswood, Memphis, TN 38152, USA