The saturation function of complete partite graphs

On Saturated $k$-Sperner Systems

“…Minimum saturation has been studied extensively in the context of graphs [1,2,5,10,12,13,18,19,20] and hypergraphs [7,14,15,16]. Such problems are typically of the following form: for a fixed (hyper)graph H, determine the minimum size of a (hyper)graph G on n vertices which does not contain a copy of H and for which adding any edge e / ∈ G, yields a (hyper)graph which contains a copy of H. This line of research was first initiated by Zykov [21] and Erdős, Hajnal and Moon [8].…”

All minimum C5-saturated graphs

“…Gould and Schmitt [14] conjectured that the complete r-partite graph K 2,...,2 has sat(n, K 2,...,2 ) = ⌈((4r − 5)n − 4r 2 + 6r − 1)/2⌉ and proved it when the minimum degree of the K 2,...,2 -saturated graphs is 2r − 3. Recently, Bohman, Fonoberova, and Pikhurko [3] proved that for r ≥ 2 and s r ≥ ... ≥ s 1 ≥ 1, as n → ∞, sat(n, K s 1 ,...,s r ) = (s 1 + ... + s r−1 + 0.5s r − 1.5)n + O(n 3/4 ). They [3] constructed a K s 1 ,...,s r -saturated graph K p * H with (s 1 + ... + s r−1 + 0.5s r − 1.5)n + O(1) edges, where H is a K 1,s r -saturated graph and p = s 1 + ... + s r−1 − 1.…”

“…Recently, Bohman, Fonoberova, and Pikhurko [3] proved that for r ≥ 2 and s r ≥ ... ≥ s 1 ≥ 1, as n → ∞, sat(n, K s 1 ,...,s r ) = (s 1 + ... + s r−1 + 0.5s r − 1.5)n + O(n 3/4 ). They [3] constructed a K s 1 ,...,s r -saturated graph K p * H with (s 1 + ... + s r−1 + 0.5s r − 1.5)n + O(1) edges, where H is a K 1,s r -saturated graph and p = s 1 + ... + s r−1 − 1. They [3] showed that any K s 1 ,...,s r -saturated graph on n vertices with at most sat(n, K s 1 ,...,s r ) + o(n) edges can be transformed into K p * H by adding and removing at most o(n) edges.…”

“…They [3] constructed a K s 1 ,...,s r -saturated graph K p * H with (s 1 + ... + s r−1 + 0.5s r − 1.5)n + O(1) edges, where H is a K 1,s r -saturated graph and p = s 1 + ... + s r−1 − 1. They [3] showed that any K s 1 ,...,s r -saturated graph on n vertices with at most sat(n, K s 1 ,...,s r ) + o(n) edges can be transformed into K p * H by adding and removing at most o(n) edges. Bohman, Fonoberova, and Pikhurko [3] also conjectured that sat(n, K 2,3 ) = 2n − 3.…”

Cycle-Saturated Graphs with Minimum Number of Edges

“…Remarkable asymptotics were given by Alon, Erdős, Holzman, and Krivelevich , (saturation and degrees). Bohman, Fonoberova, and Pikhurko determined the sat‐function asymptotically for a class of complete multipartite graphs. More recently, for multiple copies of , Faudree, Ferrara, Gould, and Jacobson determined for .…”