Related papers: The Erd\H{o}s-Hajnal property for graphs with no f…

Kelly, Kuehn and Osthus conjectured that for any l>3 and the smallest number k>2 that does not divide l, any large enough oriented graph G with minimum indegree and minimum outdegree at least \lfloor |V(G)|/k\rfloor +1 contains a directed…

A geometric graph is a graph drawn in the plane with vertices represented by points and edges as straight-line segments. A geometric graph contains a (k,l)-crossing family if there is a pair of edge subsets E_1,E_2 such that |E_1| = k and…

One of the fundamental results in graph minor theory is that for every planar graph $H$, there is a minimum integer $f(H)$ such that graphs with no minor isomorphic to $H$ have treewidth at most $f(H)$. A lower bound for ${f(H)}$ can be…

For a positive integer $k$, we say that a graph is $k$-existentially complete if for every $0 \leq a \leq k$, and every tuple of distinct vertices $x_1,\ldots,x_a$, $y_1,\ldots,y_{k-a}$, there exists a vertex $z$ that is joined to all of…

In 1977, Erd\H{o}s and Hajnal made the conjecture that, for every graph $H$, there exists $c>0$ such that every $H$-free graph $G$ has a clique or stable set of size at least $|G|^c$; and they proved that this is true with $ |G|^c$ replaced…

For every $r \in \mathbb{N}$, let $\theta_r$ denote the graph with two vertices and $r$ parallel edges. The $\theta_r$-girth of a graph $G$ is the minimum number of edges of a subgraph of $G$ that can be contracted to $\theta_r$. This…

Consider the random subgraph process on a base graph $G$ on $n$ vertices: a sequence $\lbrace G_t \rbrace _{t=0} ^{|E(G)|}$ of random subgraphs of $G$ obtained by choosing an ordering of the edges of $G$ uniformly at random, and by…

We study the appearance of powers of Hamilton cycles in pseudorandom graphs, using the following comparatively weak pseudorandomness notion. A graph $G$ is $(\varepsilon,p,k,\ell)$-pseudorandom if for all disjoint $X$ and $Y\subset V(G)$…

Let $k$ be a positive integer. Bermond and Thomassen conjectured in 1981 that every digraph with minimum outdegree at least $2k-1$ contains $k$ vertex-disjoint cycles. It is famous as one of the one hundred unsolved problems selected in…

Let $H$ be a graph with maximum degree $d$, and let $d'\ge 0$. We show that for some $c>0$ depending on $H,d'$, and all integers $n\ge 0$, there are at most $c^n$ unlabelled simple $d$-connected $n$-vertex graphs with maximum degree at most…

The famous P\'{o}sa-Seymour conjecture, confirmed in 1998 by Koml\'{o}s, S\'{a}rk\"{o}zy, and Szemer\'{e}di, states that for any $k \geq 2$, every graph on $n$ vertices with minimum degree $kn/(k + 1)$ contains the $k$-th power of a…

The bipartite-hole-number of a graph $G$, denoted by $\widetilde{\alpha}(G)$, is the minimum number $k$ such that there exist positive integers $s$ and $t$ with $s+t=k+1$ with the property that for any two disjoint sets $A,B\subseteq V(G)$…

In this short note we prove that for every $k\in \mathbb{N}$ there is a $t_k\in\mathbb{N}$ such that for every digraph $G$ there are either $k$ edge-disjoint directed cycles in $G$ or a set $X$ of at most $t_k$ edges such that $G-X$…

A conjecture of Verstra\"ete states that for any fixed $\ell < k$ there exists a positive constant $c$ such that any $C_{2k}$-free graph $G$ contains a $C_{2\ell}$-free subgraph with at least $c |E(G)|$ edges. For $\ell = 2$, this…

A monotone cylindrical graph is a topological graph drawn on an open cylinder with an infinite vertical axis satisfying the condition that every vertical line intersects every edge at most once. It is called simple if any pair of its edges…

Let $G$ be a graph, and $v\in V(G)$ and $S\subseteq V(G)\backslash v$ of size at least $k$. An important result on graph connectivity due to Perfect states that, if $v$ and $S$ are $k$-linked, then a $(k-1)$-link between a vertex $v$ and…

Erd\H{o}s and Rothschild asked to estimate the maximum number, denoted by H(N,C), such that every N-vertex graph with at least CN^2 edges, each of which is contained in at least one triangle, must contain an edge that is in at least H(N,C)…

For a nondegenerate $r$-graph $F$, large $n$, and $t$ in the regime $[0, c_{F} n]$, where $c_F>0$ is a constant depending only on $F$, we present a general approach for determining the maximum number of edges in an $n$-vertex $r$-graph that…

We prove that for every $\epsilon>0$ there exists $\delta>0$ such that the following holds. Let $\mathcal{C}$ be a collection of $n$ curves in the plane such that there are at most $(\frac{1}{4}-\epsilon)\frac{n^{2}}{2}$ pairs of curves…

Let $G$ be a simple graph with maximum degree $\Delta(G)$. A subgraph $H$ of $G$ is overfull if $|E(H)|>\Delta(G)\lfloor \frac{1}{2}|V(H)| \rfloor$. Chetwynd and Hilton in 1986 conjectured that a graph $G$ with $\Delta(G)>\frac{1}{3}|V(G)|$…