Related papers: Saturation Numbers for Berge Cliques

A graph $G$ is said to be $F$-free, if $G$ does not contain any copy of $F$. $G$ is said to be $F$-semi-saturated, if the addition of any nonedge $e \not \in E(G)$ would create a new copy of $F$ in $G+e$. $G$ is said to be $F$-saturated, if…

Let $H$ be a fixed graph. We say that a graph $G$ is $H$-saturated if it has no subgraph isomorphic to $H$, but the addition of any edge to $G$ results in an $H$-subgraph. The saturation number $\mathrm{sat}(H,n)$ is the minimum number of…

Graph $G$ is $H$-saturated if $H$ is not a subgraph of $G$ and $H$ is a subgraph of $G+e$ for any edge $e$ not in $G$. The saturation number for a graph $H$ is the minimal number of edges in any $H$-saturated graph of order $n$. In this…

A graph $H^{\prime}$ is $(H, G)$-saturated if it is $G$-free and the addition of any edge of $H$ not in $H^{\prime}$ creates a copy of $G$. The saturation number $sat(H, G)$ is the minimum number of edges in a $(H, G)$-saturated graph. We…

A graph is $F$-saturated if it is $F$-free but the addition of any edge creates a copy of $F$. In this paper we study the quantity $\mathrm{sat}(n, H, F)$ which denotes the minimum number of copies of $H$ that an $F$-saturated graph on $n$…

Given a family of graphs $\mathcal{F}$, a graph $G$ is said to be $\mathcal{F}$-saturated if $G$ does not contain a copy of $F$ as a subgraph for any $F\in\mathcal{F}$ but the addition of any edge $e\notin E(G)$ creates at least one copy of…

A graph $G$ is $F$-saturated if it does not contain any copy of $F$, but the addition of any missing edge in $G$ creates at least one copy of $F$. Inspired by work of Alon and Shikhelman regarding a similar question for $F$-free graphs,…

The linear saturation number $sat^{lin}_k(n,\mathcal{F})$ (linear extremal number $ex^{lin}_k(n,\mathcal{F})$) of $\mathcal{F}$ is the minimum (maximum) number of hyperedges of an $n$-vertex linear $k$-uniform hypergraph containing no…

A graph $G$ is $H$-saturated if it contains no $H$ as a subgraph, but does contain $H$ after the addition of any edge in the complement of $G$. The saturation number, $sat (n, H)$, is the minimum number of edges of a graph in the set of all…

Graph $G$ is $F$-saturated if $G$ contains no copy of graph $F$ but any edge added to $G$ produces at least one copy of $F$. One common variant of saturation is to remove the former restriction: $G$ is $F$-semi-saturated if any edge added…

For a given graph $F$, a graph $G$ is said to be $F$-saturated if $G$ contains no copy of $F$ but for any edge $uv\notin E(G)$, $G+uv$ contains a copy of $F$. The saturation number $sat(n,F)$ is defined as the minimum number of edges among…

The saturation number $\text{sat}(n,\mathcal{F})$ is the minimum number of edges in any graph which does not contain a member of $\mathcal{F}$ as a subgraph, but will if any edge is added. We give a few upper and lower bounds for saturation…

Let $H$ be a fixed graph, a graph G is $H$-saturated if it has no copy of $H$ in $G$, but the addition of any edge in $E(\overline G)$ to $G$ results in an $H$-subgraph. The saturation number sat$(n,H)$ is the minimum number of edges in an…

Given positive integer $n$ and graph $F$, the saturation number $\mathrm{sat}(n, F)$ is the minimum number of edges in an edge-maximal $F$-free graph on $n$ vertices. In this paper, we determine asymptotic behavior of $\mathrm{sat}(n, F)$…

Given a family ${\mathcal F}$ and a host graph $H$, a graph $G\subseteq H$ is ${\mathcal F}$-saturated relative to $H$ if no subgraph of $G$ lies in ${\mathcal F}$ but adding any edge from $E(H)-E(G)$ to $G$ creates such a subgraph. In the…

Let $K^r_n$ be the complete $r$-uniform hypergraph on $n$ vertices, that is, the hypergraph whose vertex set is $[n]:=\{1,2,...,n\}$ and whose edge set is $\binom{[n]}{r}$. We form $G^r(n,p)$ by retaining each edge of $K^r_n$ independently…

Given a family of graphs $\mathcal{F}$, a graph $G$ is said to be $\mathcal{F}$-saturated if $G$ does not contain a copy of $F$ as a subgraph for any $F\in\mathcal{F}$, but the addition of any edge $e\notin E(G)$ creates at least one copy…

For graphs $G$ and $F$, the saturation number $\textit{sat}(G,F)$ is the minimum number of edges in an inclusion-maximal $F$-free subgraph of $G$. In 2017, Kor\'andi and Sudakov initiated the study of saturation in random graphs. They…

For a given graph $F$, the $F$-saturation number of a graph $G$, denoted by $ {sat}(G, F)$, is the minimum number of edges in an edge-maximal $F$-free subgraph of $G$. In 2017, Kor\'andi and Sudakov determined $ {sat}({G}(n, p), K_r)$…

The saturation number of a graph $F$, written $\textup{sat}(n,F)$, is the minimum number of edges in an $n$-vertex $F$-saturated graph. One of the earliest results on saturation numbers is due to Erd\H{o}s, Hajnal, and Moon who determined…