Related papers: Some results on minimum saturated graphs

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…

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…

An edge-coloring of a graph $H$ is a function $\mathcal{C}: E(H) \rightarrow \mathbb{N}$. We say that $H$ is rainbow if all edges of $H$ have different colors. Given a graph $F$, an edge-colored graph $G$ is $F$-rainbow saturated if $G$…

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…

We investigate the extremal properties of saturated partial plane embeddings of maximal planar graphs. For a planar graph $G$, the plane-saturation number $\mathrm{sat}_{\mathcal{P}}(G)$ denotes the minimum number of edges in a plane…

A graph $G = (V,E)$ is said to be saturated with respect to a monotone increasing graph property ${\mathcal P}$, if $G \notin {\mathcal P}$ but $G \cup \{e\} \in {\mathcal P}$ for every $e \in \binom{V}{2} \setminus E$. The saturation game…

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…

We investigate two notions of saturation for partial planar embeddings of maximal planar graphs. Let $G = (V, E) $ be a vertex-labeled maximal planar graph on $ n $ vertices, which by definition has $3n - 6$ edges. We say that a labeled…

A simple topological graph $G$ is a graph drawn in the plane so that any pair of edges have at most one point in common, which is either an endpoint or a proper crossing. $G$ is called saturated if no further edge can be added without…

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

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…

For a graph $H$, a graph $G$ is $H$-saturated if $G$ does not contain $H$ as a subgraph but for any $e \in E(\overline{G})$, $G+e$ contains $H$. In this note, we prove a sharp lower bound for the number of paths and walks on length $2$ in…

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…

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…

We study the problem of determining $sat(n,k,r)$, the minimum number of edges in a $k$-partite graph $G$ with $n$ vertices in each part such that $G$ is $K_r$-free but the addition of an edge joining any two non-adjacent vertices from…

Given a graph $F$, a graph $G$ is weakly $F$-saturated if all non-edges of $G$ can be added in some order so that each new edge introduces a copy of $F$. The weak saturation number $\operatorname{wsat}(n, F)$ is the minimum number of edges…

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…

Given a graph $G$, we say a $k$-uniform hypergraph $H$ on the same vertex set contains a Berge-$G$ if there exists an injection $\phi:E(G)\to E(H)$ such that $e\subseteq\phi(e)$ for each edge $e\in E(G)$. A hypergraph $H$ is…

For a graph G, a hypergraph H is called Berge-G if there is a hypergraph H', isomorphic to H, containing all vertices of G, so that e is contained in f(e) for each edge e of G, where f is a bijection between E(G) and E(H'). The set of all…