Related papers: The minimum spectral radius of $tP_4$-saturated gr…

Let $G$ be a $K_4$-free graph, an edge in its complement is a $K_4$-\emph{saturating} edge if the addition of this edge to $G$ creates a copy of $K_4$. Erd\H{o}s and Tuza conjectured that for any $n$-vertex $K_4$-free graph $G$ with…

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…

Write $\rho\left( G\right) $ for the spectral radius of a graph $G$ and $S_{n,r}$ for the join $K_{r}\vee\overline{K}_{n-r}.$ Let $n>r\geq2$ and $G$ be a $K_{r+1}$-saturated graph of order $n.$ Recently Kim, Kim, Kostochka, and O determined…

For two given graphs $G$ and $F$, a graph $ H$ is said to be weakly $ (G, F) $-saturated if $H$ is a spanning subgraph of $ G$ which has no copy of $F$ as a subgraph and one can add all edges in $ E(G)\setminus E(H)$ to $ H$ in some order…

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…

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 $F$-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…

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 $n$,…

Given graphs $H$ and $F$, a subgraph $G\subseteq H$ is an $F$-saturated subgraph of $H$ if $F\nsubseteq G$, but $F\subseteq G+e$ for all $e\in E(H)\setminus E(G)$. The saturation number of $F$ in $H$, denoted $\text{sat}(H,F)$, is the…

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.…

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…

Given graphs $H_1, \dots, H_t$, a graph $G$ is $(H_1, \dots, H_t)$-Ramsey-minimal if every $t$-coloring of the edges of $G$ contains a monochromatic $H_i$ in color $i$ for some $i\in\{1, \dots, t\}$, but any proper subgraph of $G $ does not…

Let $F$ be a graph and $\mathcal{H}$ be a hypergraph, both embedded on the same vertex set. We say $\mathcal{H}$ is a Berge-$F$ if there exists a bijection $\phi:E(F)\to E(\mathcal{H})$ such that $e\subseteq \phi(e)$ for all $e\in E(F)$. We…

Given a graph $F$, a hypergraph is a Berge-$F$ if it can be obtained by expanding each edge in $F$ to a hyperedge containing it. A hypergraph $H$ is Berge-$F$-saturated if $H$ does not contain a subgraph that is a Berge-$F$, but for any…

A graph $H$ is $K_s$-saturated if it is a maximal $K_s$-free graph, i.e., $H$ contains no clique on $s$ vertices, but the addition of any missing edge creates one. The minimum number of edges in a $K_s$-saturated graph was determined over…

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…

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 $F$ and $G$ be two graphs. A spanning subgraph $H$ of $G$ is called weakly $F$-saturated if one can add to $H$ the edges of $G \setminus H$ in some order, so that whenever a new edge is added, a new copy of $F$ is formed. Obtaining…

Given an integer $r\ge1$ and graphs $G, H_1, \ldots, H_r$, we write $G \rightarrow ({H}_1, \ldots, {H}_r)$ if every $r$-coloring of the edges of $G$ contains a monochromatic copy of $H_i$ in color $i$ for some $i\in\{1, \ldots, r\}$. A…

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…