Related papers: Partite Saturation of Complete Graphs

A graph $G$ is $k$-critical if it has chromatic number $k$, but every proper subgraph of $G$ is $(k-1)$--colorable. Let $f_k(n)$ denote the minimum number of edges in an $n$-vertex $k$-critical graph. We give a lower bound, $f_k(n) \geq…

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…

Let $\mathcal{D}$ be a family of digraphs. A digraph $D$ is \emph{$\mathcal{D}$-saturated} if it contains no member of $\mathcal{D}$ as a subdigraph, but for any arc $e$ in the complement of $D$, the digraph $D + e$ contains some member of…

Our goal is to investigate a close relative of the independent transversal problem in the class of infinite $K_n$-free graphs: we show that for any infinite $K_n$-free graph $G=(V,E)$ and $m\in \mathbb N$ there is a minimal $r=r(G,m)$ such…

Popielarz, Sahasrabudhe and Snyder in 2018 proved that maximal $K_{r+1}$-free graphs with $(1-\frac{1}{r})\frac{n^2}{2}-o(n^{\frac{r+1}{r}})$ edges contain a complete $r$-partite subgraph on $n-o(n)$ vertices. This was very recently…

Let $H$ be a fixed graph, a graph G is $H$-saturated if it has no copy of $H$ in $G$, but the addition of any edge in $E(\overline G)$ to $G$ results in an $H$-subgraph. The saturation number sat$(n,H)$ is the minimum number of edges in an…

Given $r\in \mathbb{N}$ with $r\geq 4$, we show that there exists $n_0\in \mathbb{N}$ such that for every $n\geq n_0$, every $n$-vertex graph $G$ with $\delta(G)\geq (\frac{1}{2}+o(1))n$ and $\alpha_{r-2}(G)=o(n)$ contains a $K_{r}$-factor.…

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…

We call an edge-colored graph rainbow if all of its edges receive distinct colors. An edge-colored graph $\Gamma$ is called $H$-rainbow saturated if $\Gamma$ does not contain a rainbow copy of $H$ and adding an edge of any color to $\Gamma$…

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…

Given a family $\mathcal{F}$ of bipartite graphs, the {\it Zarankiewicz number} $z(m,n,\mathcal{F})$ is the maximum number of edges in an $m$ by $n$ bipartite graph $G$ that does not contain any member of $\mathcal{F}$ as a subgraph (such…

Given graphs $H_1, H_2$, a {red, blue}-coloring of the edges of a graph $G$ is a critical coloring if $G$ has neither a red $H_1$ nor a blue $ H_2$. A non-complete graph $G$ is $(H_1, H_2)$-co-critical if $G$ admits a critical coloring, but…

We show that for each $k\geq 4$ and $n>r\geq k+1$, every $n$-vertex $r$-uniform hypergraph with no Berge cycle of length at least $k$ has at most $\frac{(k-1)(n-1)}{r}$ edges. The bound is exact, and we describe the extremal hypergraphs.…

A central problem in extremal graph theory is to estimate, for a given graph $H$, the number of $H$-free graphs on a given set of $n$ vertices. In the case when $H$ is not bipartite, fairly precise estimates on this number are known. In…

Given a graph $H$, the extremal number $\mathrm{ex}(n,H)$ is the largest number of edges in an $H$-free graph on $n$ vertices. We make progress on a number of conjectures about the extremal number of bipartite graphs. First, writing…

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…

We establish the following strengthening of the celebrated Andr{\'a}sfai--Erd\H{o}s--S\'{o}s theorem: If $G$ is an $n$-vertex $K_{r+1}$-free graph whose minimum degree $\delta(G)$ and maximum degree $\Delta(G)$ satisfy \begin{align*}…

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…

A well known result due to Andr\'asfai, Erd\H{o}s, and S\'os asserts that for $r\ge 2$ every $K_{r+1}$-free graph on $n$ vertices with $\delta(G)>\frac{3r-4}{3r-1}n$ is $r$-partite. We study related questions in the context of weighted…

Given two graphs $G, H$ and a positive integer $q$, an $(H,q)$-coloring of $G$ is an edge-coloring of $G$ such that every copy of $H$ in $G$ receives at least $q$ distinct colors. The bipartite Erd\H{o}s-Gy\'{a}rf\'{a}s function $r(K_{n,n},…