Related papers: $K_{s,t}$-saturated bipartite graphs

In 1964, Erd\H{o}s, Hajnal and Moon introduced a saturation version of Tur\'an's classical theorem in extremal graph theory. In particular, they determined the minimum number of edges in a $K_r$-free, $n$-vertex graph with the property that…

For a class $\mathcal{D}$ of drawings of loopless (multi-)graphs in the plane, a drawing $D \in \mathcal{D}$ is \emph{saturated} when the addition of any edge to $D$ results in $D' \notin \mathcal{D}$ - this is analogous to saturated graphs…

An $n$-vertex graph $G$ is weakly $F$-saturated if $G$ contains no copy of $F$ and there exists an ordering of all edges in $E(K_n) \setminus E(G)$ such that, when added one at a time, each edge creates a new copy of $F$. The minimum size…

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…

We look at several saturation problems in complete balanced blow-ups of graphs. We let $H[n]$ denote the blow-up of $H$ onto parts of size $n$ and refer to a copy of $H$ in $H[n]$ as 'partite' if it has one vertex in each part of $H[n]$. We…

The problem of Zarankiewicz asks for the maximum number of edges in a bipartite graph on $n$ vertices which does not contain the complete bipartite graph $K_{k,k}$ as a subgraph. A classical theorem due to K\H{o}v\'ari, S\'os, and Tur\'an…

An $\mathcal{F}$-saturated $r$-graph is a maximal $r$-graph not containing any member of $\mathcal{F}$ as a subgraph. Let $\mathcal{K}_{\ell + 1}^{r}$ be the collection of all $r$-graphs $F$ with at most $\binom{\ell+1}{2}$ edges such that…

We show tight necessary and sufficient conditions on the sizes of small bipartite graphs whose union is a larger bipartite graph that has no large bipartite independent set. Our main result is a common generalization of two classical…

In 1984, Erd\H{o}s and Simonovits conjectured the following: given a bipartite graph $H$, there exist constants $\beta, C > 0$ such that any graph $G$ on $n$ vertices and $pn^2\geq C \mathrm{ex}(n, H)$ edges contains at least $\beta…

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…

Let $f_k(n,H)$ denote the maximum number of edges not contained in any monochromatic copy of~$H$ in a $k$-coloring of the edges of $K_n$, and let $ex(n,H)$ denote the Tur\'an number of $H$. In place of $f_2(n,H)$ we simply write $f(n,H)$.…

For an odd integer $k$, let $\mathcal{C}_k = \{C_3,C_5,...,C_k\}$ denote the family of all odd cycles of length at most $k$ and let $\mathcal{C}$ denote the family of all odd cycles. Erd\H{o}s and Simonovits \cite{ESi1} conjectured that for…

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…

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 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…

The saturation number of a graph $F$, written $\textup{sat}(n,F)$, is the minimum number of edges in an $n$-vertex $F$-saturated graph. One of the earliest results on saturation numbers is due to Erd\H{o}s, Hajnal, and Moon who determined…

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…

A graph $H^{\prime}$ is $(H, G)$-saturated if it is $G$-free and the addition of any edge of $H$ not in $H^{\prime}$ creates a copy of $G$. The saturation number $sat(H, G)$ is the minimum number of edges in a $(H, G)$-saturated graph. We…

Given a graph $H$, we say that an edge-coloured graph $G$ is $H$-rainbow saturated if it does not contain a rainbow copy of $H$, but the addition of any non-edge in any colour creates a rainbow copy of $H$. The rainbow saturation number…

Let $k \ge 2$ be an integer. We show that if $s = 2$ and $t \ge 2$, or $s = t = 3$, then the maximum possible number of edges in a $C_{2k+1}$-free graph containing no induced copy of $K_{s,t}$ is asymptotically equal to $(t - s +…