Saturation numbers for families of Ramsey-minimal graphs

Abstract

Given a family of graphs F , a graph G is F-saturated if no element of F is a subgraph of G, but for any edge e in G, some element of F is a subgraph of G + e. Let sat(n,F) denote the minimum number of edges in an F -saturated graph of order n. For graphs G,H1, . . . , Hk, we write that G → (H1, . . . , Hk) if every k-coloring of E(G) contains a monochromatic copy of Hi in color i for some i. A graph G is (H1, . . . , Hk)-Ramsey-minimal if G → (H1, . . . , Hk) but for any e ∈ G, (G − e) 6→ (H1, . . . , Hk). Let Rmin(H1, . . . , Hk) denote the family of (H1, . . . , Hk)-Ramseyminimal graphs. In 1987, Hanson and Toft conjectured that