Related papers: Partite Saturation of Complete Graphs

A graph $G$ is called $F$-saturated if $G$ does not contain $F$ as a subgraph (not necessarily induced) but the addition of any missing edge to $G$ creates a copy of $F$. The saturation number of $F$, denoted by $sat(n,F)$, is the minimum…

A graph $H$ is said to be $F$-saturated relative to $G$, if $H$ does not contain any copy of $F$, but the addition of any edge $e$ in $E(G)\backslash E(H)$ would create a copy of $F$. The minimum size of an $F$-saturated graph relative to…

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)$…

We say that two vertices are twins if they have the same neighbourhood and that a graph is $K_r$-saturated if it does not contain $K_r$ but adding any new edge to it creates a $K_r$. In 1964, Erd\H{o}s, Hajnal and Moon showed that…

Given a graph $H$, we say that a graph $G$ is $H$-saturated if $G$ contains no copy of $H$ but adding any new edge to $G$ creates a copy of $H$. Let $sat(n,K_r,t)$ be the minimum number of edges in a $K_r$-saturated graph on $n$ vertices…

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…

Let $\mathcal{F}$ be a family of $r$-graphs. An $r$-graph $G$ is called $\mathcal{F}$-saturated if it does not contain any members of $\mathcal{F}$ but adding any edge creates a copy of some $r$-graph in $\mathcal{F}$. The saturation number…

A graph $G$ is called $C_k$-saturated if $G$ is $C_k$-free but $G+e$ not for any $e\in E(\overline{G})$. The saturation number of $C_k$, denoted $sat(n,C_k)$, is the minimum number of edges in a $C_k$-saturated graph on $n$ vertices.…

Let $G$ be a graph and $\mathcal{F}$ be a family of graphs. We say a graph $G$ is $\mathcal{F}$-saturated if $G$ does not contain any member in $\mathcal{F}$ and for any $e\in E(\overline{G})$, $G+e$ creates a copy of some member in $…

Let $G$ be a fixed graph and let ${\mathcal F}$ be a family of graphs. A subgraph $J$ of $G$ is ${\mathcal F}$-saturated if no member of ${\mathcal F}$ is a subgraph of $J$, but for any edge $e$ in $E(G)-E(J)$, some element of ${\mathcal…

A graph $G$ is $H$-saturated if it contains no copy of $H$ as a subgraph but the addition of any new edge to $G$ creates a copy of $H$. In this paper we are interested in the function sat$_{t}(n,p)$, defined to be the minimum number of…

An $n$-by-$n$ bipartite graph is $H$-saturated if the addition of any missing edge between its two parts creates a new copy of $H$. In 1964, Erd\H{o}s, Hajnal and Moon made a conjecture on the minimum number of edges in a…

The saturation number $\operatorname{sat}(n, H)$ of a graph $H$ and positive integer $n$ is the minimum size of a graph of order $n$ which does not contain a subgraph isomorphic to $H$ but to which the addition of any edge creates such a…

A graph $G$ is called $H$-saturated if $G$ contains no copy of $H$, but $G+e$ contains a copy of $H$ for any edge $e\in E(\overline{G})$. The saturation number of $H$ is the minimum number of edges in an $H$-saturated graph of order $n$,…

Let $\mathcal{F}$ be a family of graphs. A graph $G$ is $\mathcal{F}$-saturated if $G$ contains no member of $\mathcal{F}$ as a subgraph but $G+e$ contains some member of $\mathcal{F}$ whenever $e\in E(\overline{G})$. The saturation number…

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…

A graph $G$ is $F$-saturated if $G$ is $F$-free but for any edge $e$ in the complement of $G$ the graph $G + e$ contains $F$. Gerbner et al. (Discrete Math., 345 (2022), 112921) initiated the study of $rsat(n,F)$, the minimum number of…

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…

The saturation number $\text{sat}_r(n,\mathcal{F})$ is the minimum number of hyperedges in an $r$-uniform $\mathcal{F}$-saturated hypergraph on $n$ vertices. We determine this parameter for $3$-uniform Berge-$K_4$ hypergraphs, proving that…

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 $\mathrm{sat}(n, F)$…