Related papers: Minimum saturated graphs without $4$-cycles and $5…

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 $n$,…

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…

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…

We say $G$ is \emph{$(Q_n,Q_m)$-saturated} if it is a maximal $Q_m$-free subgraph of the $n$-dimensional hypercube $Q_n$. A graph, $G$, is said to be $(Q_n,Q_m)$-semi-saturated if it is a subgraph of $Q_n$ and adding any edge forms a new…

We say that an edge-coloring of a graph $G$ is proper if every pair of incident edges receive distinct colors, and is rainbow if no two edges of $G$ receive the same color. Furthermore, given a fixed graph $F$, we say that $G$ is rainbow…

Gerbner, Patk\'{o}s, Tuza, and Vizer recently initiated the study of $F$-saturated regular graphs. One of the essential problems in this line of research is determining when such a graph exists. Using generalized sum-free sets we prove that…

Let $\mathcal{C}$ be a family of edge-colored graphs. A $t$-edge colored graph $G$ is $(\mathcal{C}, t)$-saturated if $G$ does not contain any graph in $\mathcal{C}$ but the addition of any edge in any color in $[t]$ creates a copy of some…

Motivated by the work of Anstee, Griggs, and Sali on forbidden submatrices and the extremal sat-function for graphs, we introduce sat-type problems for matrices. Let F be a family of k-row matrices. A matrix M is called F-admissible if M…

Given a family of graphs $\mathcal{F}$, we consider the $\mathcal{F}$-saturation game. In this game two players alternate adding edges to an initially empty graph on $n$ vertices, with the only constraint being that neither player can add…

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…

Given a family of graphs $\mathcal{F}$, a graph $G$ is $\mathcal{F}$-saturated if it is $\mathcal{F}$-free but the addition of any missing edge creates a copy of some $F \in \mathcal{F}$. The study of the minimum number of edges in…

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…

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)$…

A graph $G$ is $F$-saturated if it contains no copy of $F$ as a subgraph but the addition of any new edge to $G$ creates a copy of $F$. We prove that for $s \geq 3$ and $t \geq 2$, the minimum number of copies of $K_{1,t}$ in a…

For a graph $F$, we say a hypergraph $H$ is Berge-$F$ if it can be obtained from $F$ be replacing each edge of $F$ with a hyperedge containing it. We say a hypergraph is Berge-$F$-saturated if it does not contain a Berge-$F$, but adding any…

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…

For a given graph $F$, the $F$-saturation number of a graph $G$ is the minimum number of edges in an edge-maximal $F$-free subgraph of $G$. Recently, the $F$-saturation number of the Erd\H{o}s$\text{\bf--}$R\'enyi random graph…

The saturation number of a graph $G$ is the cardinality of any smallest maximal matching of $G$, and it is denoted by $s(G)$. Fullerene graphs are cubic planar graphs with exactly twelve 5-faces; all the other faces are hexagons. They are…

A hypergraph $H=(V(H), E(H))$ is a Berge copy of a graph $F$, if $V(F)\subset V(H)$ and there is a bijection $f:E(F)\rightarrow E(H)$ such that for any $e\in E(F)$ we have $e\subset f(e)$. A hypergraph is Berge-$F$-free if it does not…