Related papers: Saturation Numbers for Linear Forests $P_6$ + $tP_…

For a graph $H$, a graph $G$ is $H$-induced-saturated if $G$ does not contain an induced copy of $H$, but either removing an edge from $G$ or adding a non-edge to $G$ creates an induced copy of $H$. Depending on the graph $H$, an…

A drawing of a graph is $k$-plane if every edge contains at most $k$ crossings. A $k$-plane drawing is saturated if we cannot add any edge so that the drawing remains $k$-plane. It is well-known that saturated $0$-plane drawings, that is,…

A graph $G$ is called $H$-induced-saturated if $G$ does not contain an induced copy of $H$, but removing any edge from $G$ creates an induced copy of $H$ and adding any edge of $G^{c}$ to $G$ creates an induced copy of $H$. Martin and Smith…

Given an $r$-uniform hypergraph $H$ and a positive integer $n$, the weak saturation number $\mathrm{wsat}(n,H)$ is the minimum number of edges in an $r$-uniform hypergraph $F$ on $n$ vertices such that the missing edges in $F$ can be added,…

For two graphs $J$ and $H$, the generalized Tur\'{a}n number, denoted by $ex(n,J,H)$, is the maximum number of copies of $J$ in an $H$-free graph of order $n$. A linear forest $F$ is the disjoint union of paths. In this paper, we determine…

In this paper, we address problems related to parameters concerning edge mappings of graphs. The quantity $h(n,G)$ is defined to be the maximum number of edges in an $n$-vertex graph $H$ such that there exists a mapping $f: E(H)\rightarrow…

The process of $H$-bootstrap percolation for a graph $H$ is a cellular automaton, where, given a subset of the edges of $K_n$ as initial set, an edge is added at time $t$ if it is the only missing edge in a copy of $H$ in the graph obtained…

The paper studies wsat$(G,H)$ which is the minimum number of edges in a weakly $H$-saturated subgraph of $G$. We prove that wsat$(K_n,H)$ is `stable' - remains the same after independent removal of every edge of $K_n$ with constant…

A degree monotone path in a graph $G$ is a path $P$ such that the sequence of degrees of the vertices in the order in which they appear on $P$ is monotonic. The length of the longest degree monotone path in $G$ is denoted by $mp(G)$. This…

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…

The Tur\'an number of a graph $H$, denoted by $ex(n,H)$, is the maximum number of edges in any graph on $n$ vertices containing no $H$ as a subgraph. A linear (star) forest is a forest consisting of paths (stars). A path-star forest $F$ is…

Let $G=(V,E)$ be a simple connected graph. A matching $M$ in a graph $G$ is a collection of edges of $G$ such that no two edges from $M$ share a vertex. A matching $M$ is maximal if it cannot be extended to a larger matching in $G$. The…

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 finite poset $\mathcal P$, how small can a family $\mathcal F$ of subsets of $[n]$ be such that $\mathcal F$ does not contain an induced copy of $\mathcal P$, but $\mathcal F\cup\{X\}$ contains such a copy for all $X\in\mathcal…

A simple topological graph is a topological graph in which any two edges have at most one common point, which is either their common endpoint or a proper crossing. More generally, in a k-simple topological graph, every pair of edges has at…

A graph $G$ is $H$-induced-saturated if $G$ is $H$-free but deleting any edge or adding any edge creates an induced copy of $H$. There are non-trivial graphs $H$, such as $P_4$, for which no finite $H$-induced-saturated graph $G$ exists. We…

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…

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 graphs $G$ and $H$, we say that $G$ is $H$-free if no induced subgraph of $G$ is isomorphic to $H$, and that $G$ is $H$-induced-saturated if $G$ is $H$-free but removing or adding any edge in $G$ creates an induced copy of $H$. A full…

A graph $G$ is uniquely $H$-saturated if it contains no copy of a graph $H$ as a subgraph, but adding any new edge into $G$ creates exactly one copy of $H$. Let $C_{4}^{+}$ be the diamond graph consisting of a $4$-cycle $C_{4}$ with one…