Related papers: Saturation Numbers for Berge Cliques

Given a graph $H$, we say that a graph $G$ is properly rainbow $H$-saturated if: (1) There is a proper edge colouring of $G$ containing no rainbow copy of $H$; (2) For every $e \notin E(G)$, every proper edge colouring of $G+e$ contains a…

This paper considers two important questions in the well-studied theory of graphs that are $F$-saturated. A graph $G$ is called $F$-saturated if $G$ does not contain a subgraph isomorphic to $F$, but the addition of any edge creates a copy…

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…

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…

For an edge-ordered graph $G$, we say that an $n$-vertex edge-ordered graph $H$ is $G$-saturated if it is $G$-free and adding any new edge with any new label to $H$ introduces a copy of $G$. The saturation function describes the minimum…

A graph $G$ is called $H$-saturated if it does not contain any copy of $H$, but for any edge $e$ in the complement of $G$ the graph $G+e$ contains some $H$. The minimum size of an $n$-vertex $H$-saturated graph is denoted by $\sat(n,H)$. We…

Given graphs $G$ and $H$, $G$ is $H$-saturated if $G$ does not contain a copy of $H$ but the addition of any edge $e\notin E(G)$ creates at least one copy of $H$ within $G$. The edge spectrum of $H$ is the set of all possible sizes of an…

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…

For a fixed graph $F,$ the minimum number of edges in an edge-maximal $F$-free subgraph of $G$ is called the $F$-saturation number. The asymptotics of the $F$-saturation number of the binomial random graph $G(n,p)$ for constant $p\in(0,1)$…

For a fixed graph $H$, a graph $G$ is called $H$-saturated if $G$ does not contain $H$ as a (not necessarily induced) subgraph, but $G+e$ contains a copy of $H$ for any $e\in E(\overline{G})$. The saturation number of $H$, denoted by ${\rm…

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…

Fix a hypergraph $\mathcal{F}$. A hypergraph $\mathcal{H}$ is called a {\it Berge copy of $\mathcal{F}$} or {\it Berge-$\mathcal{F}$} if we can choose a subset of each hyperedge of $\mathcal{H}$ to obtain a copy of $\mathcal{F}$. A…

A graph $G$ is $H$-saturated for a graph $H$, if $G$ does not contain a copy of $H$ but adding any new edge to $G$ results in such a copy. An $H$-saturated graph on a given number of vertices always exists and the properties of such graphs,…

Given a graph $H$, we say a graph $G$ is properly rainbow $H$-saturated if there is a proper edge-coloring of $G$ which contains no rainbow copy of $H$, but adding any edge to $G$ makes such an edge-coloring impossible. The proper rainbow…

Let $\mathcal{F}$ be an $r$-uniform hypergraph and $G$ be a multigraph. The hypergraph $\mathcal{F}$ is a Berge-$G$ if there is a bijection $f: E(G) \rightarrow E( \mathcal{F} )$ such that $e \subseteq f(e)$ for each $e \in E(G)$. Given a…

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.…

For given graphs $F$ and $G$, the minimum number of edges in an inclusion-maximal $F$-free subgraph of $G$ is called the $F$-saturation number and denoted $\mathrm{sat}(G, F)$. For the star $F=K_{1,r}$, the asymptotics of…

Given graphs $G$ and $H$, $G$ is $H$-saturated if $H$ is not a subgraph of $G$, but for all $e \notin E(G)$, $H$ appears as a subgraph of $G + e$. While for every $n \ge |V(H)|$, there exists an $n$-vertex graph that is $H$-saturated, the…

Let $F$ be a graph. We say that a hypergraph $H$ is a {\it Berge}-$F$ if there is a bijection $f : E(F) \rightarrow E(H )$ such that $e \subseteq f(e)$ for every $e \in E(F)$. Note that Berge-$F$ actually denotes a class of hypergraphs. The…

Given a hypergraph $\mathcal{H}$ and a graph $G$, we say that $\mathcal{H}$ is a \textit{Berge}-$G$ if there is a bijection between the hyperedges of $\mathcal{H}$ and the edges of $G$ such that each hyperedge contains its image. We denote…