Extremal problems on saturation for the family of k-edge-connected graphs

Abstract Let F be a family of graphs. A graph G is F -saturated if G contains no member of F as a subgraph but G + e contains some member of F whenever e ∈ E ( G ¯ ) . The saturation number and extremal number of F , denoted sat ( n , F ) and ex ( n , F ) respectively, are the minimum and maximum numbers of edges among n -vertex F -saturated graphs. For k ∈ N , let F k and F k ′ be the families of k -connected and k -edge-connected graphs, respectively. Wenger proved sat ( n , F k ) = ( k − 1 ) n − k 2 ; we prove sat ( n , F k ′ ) = ( k − 1 ) ( n − 1 ) − n k + 1 k − 1 2 . We also prove ex ( n , F k ′ ) = ( k − 1 ) n − k 2 and characterize when equality holds. Finally, we give a lower bound on the spectral radius for F k -saturated and F k ′ -saturated graphs.