Ramsey-minimal saturation numbers for matchings

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) over bar, 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, H-1, ... , H-k, we write that G -> (H-1, ... , H-k) if every k-coloring of E(G) contains a monochromatic copy of H-i in color i for some i. A graph G is (H-1, ... , H-k)-Ramsey-minimal if G -> (H-1, ... , H-k) but for any e is an element of G, (G - e) negated right arrow (H-1, ... , H-k). Let R-min (H-1, ... , H-k) denote the family of (H-1, ... , H-k)-Ramsey-minimal graphs. In this paper, motivated in part by a conjecture of Hanson and Toft (1987), we prove that sat(n, R-min(m(1)K(2), ... , m(k)K(2))) = 3(m(1) + ... + m(k) - k) for m(1), ... , m(k) >= 1 and n > 3(m(1) + ... + m(k) - 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 H-i-saturated graphs to determine the saturation number of R-min(H-1, ... , H-k). (C) 2014 Elsevier B.V. All rights reserved.