Ramsey-minimal saturation numbers for matchings

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, which we refer to as the saturation number or saturation function of F . If F = {F}, then we instead say that G is F -saturated and write sat(n, F ). 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)-Ramseyminimal if G→ (H1, . . . ,Hk) but for any e ∈ G, (G−e) 6→ (H1, . . . ,Hk). Let Rmin(H1, . . . ,Hk) denote the family of (H1, . . . ,Hk)-Ramsey-minimal graphs. In this paper, motivated in part by a conjecture of Hanson and Toft [Edge-colored saturated graphs, J. Graph Theory 11 (1987), 191–196], we prove that sat(n,Rmin(m1K2, . . . ,mkK2)) = 3(m1 + . . .+mk − k) for m1, . . . ,mk ≥ 1 and n > 3(m1 + . . .+mk−k), and we also characterize the saturated graphs of minimum size. The proof of this result uses a new technique, iterated recoloring, which takes advantage of the structure of Hi-saturated graphs to determine the saturation number of Rmin(H1, . . . ,Hk).