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 s a t ( 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 ) . aphs 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 ∈ G , ( G − e ) ⁄ → ( H 1 , … , H k ) . Let R min ( H 1 , … , H k ) denote the family of ( H 1 , … , H k ) -Ramsey-minimal graphs. s 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 ) .