Related papers: A step towards a general density Corr\'{a}di--Hajn…

As an application of Szemeredi's regularity lemma, Erdos-Frankl-Rodl (1986) showed that the number of graphs on vertex set {1,2,...n} with a monotone class P is $2^{(1+o(1))ex(n,P)n^2/2}$ where $ex(n,P)$ is the maximum number of edges of an…

In a recent paper, Hunter, Milojevi\'c, Sudakov and Tomon consider the maximum number of edges in an $n$-vertex graph containing no copy of the complete bipartite graph $K_{s,s}$ and no induced copy of a "pattern" graph $H$. They conjecture…

Hajos' conjecture that every simple even graph on $n$ vertices can be decomposed into at most $(n-1)/2$ cycles (see L. Lovasz, On covering of graphs, in: P. Erdos, G.O.H. Katona (Eds.), Theory of Graphs, Academic Press, New York, 1968, pp.…

In the 1960s, Erd\H{o}s and his cooperators initiated the research of the maximum numbers of edges in a graph or a planar graph on $n$ vertices without $k$ edge-disjoint cycles. This problem had been solved for $k\leq4$. As pointed out by…

An extremal graph for a graph $H$ on $n$ vertices is a graph on $n$ vertices with maximum number of edges that does not contain $H$ as a subgraph. Let $T_{n,r}$ be the Tur\'{a}n graph, which is the complete $r$-partite graph on $n$ vertices…

We study the following question: how few edges can we delete from any $H$-free graph on $n$ vertices in order to make the resulting graph $k$-colorable? It turns out that various classical problems in extremal graph theory are special cases…

This paper is motivated by the question of how global and dense restriction sets in results from extremal combinatorics can be replaced by less global and sparser ones. The result we consider here as an example is Turan's theorem, which…

We generalise a result of Corr\'{a}di and Hajnal and show that every graph with average degree at least $\tfrac{4}{3}kr$ contains $k$ vertex disjoint cycles, each of order at least $r$, as long as $k \geq 6$. This bound is sharp when $r=3$.

Let $\mathcal{H}=(V,\mathcal{E})$ be an $r$-uniform hypergraph on $n$ vertices and fix a positive integer $k$ such that $1\le k\le r$. A $k$-\emph{matching} of $\mathcal{H}$ is a collection of edges $\mathcal{M}\subset \mathcal{E}$ such…

The \emph{Tur\'an function} $\ex(n,F)$ of a graph $F$ is the maximum number of edges in an $F$-free graph with $n$ vertices. The classical results of Tur\'an and Rademacher from 1941 led to the study of supersaturated graphs where the key…

We systematically study a natural problem in extremal graph theory, to minimize the number of edges in a graph with a fixed number of vertices, subject to a certain local condition: each vertex must be in a copy of a fixed graph $H$. We…

Erd\H{o}s proved an upper bound on the number of edges in an $n$-vertex non-Hamiltonian graph with given minimum degree and showed sharpness via two members of a particular graph family. F\"{u}redi, Kostochka and Luo showed that these two…

Let $\mathcal{H}$ be a hypergraph and $F$ be a graph. If there exists a bijection between the hyperedges of $\mathcal{H}$ and the edges of $F$ such that each hyperedge contains its image, then we say that $\mathcal{H}$ is a \textit{Berge…

The chromatic threshold $\delta_\chi(H)$ of a graph $H$ is the infimum of $d>0$ such that the chromatic number of every $n$-vertex $H$-free graph with minimum degree at least $dn$ is bounded in terms of $H$ and $d$. A breakthrough result of…

Let $F$ be an $(r+1)$-color critical graph with $r\geq 2$, that is, $\chi(F)=r+1$ and there is an edge $e$ in $F$ such that $\chi(F-e)=r$. Gerbner recently conjectured that every $n$-vertex maximal $F$-free graph with at least…

We investigate the number of 4-edge paths in graphs with a fixed number of vertices and edges. An asymptotically sharp upper bound is given to this quantity. The extremal construction is the quasi-star or the quasi-clique graph, depending…

For graphs $G$ and $H$, an $H$-coloring of $G$ is an adjacency preserving map from the vertices of $G$ to the vertices of $H$. $H$-colorings generalize such notions as independent sets and proper colorings in graphs. There has been much…

For two graphs $T$ and $H$ with no isolated vertices and for an integer $n$, let $ex(n,T,H)$ denote the maximum possible number of copies of $T$ in an $H$-free graph on $n$ vertices. The study of this function when $T=K_2$ is a single edge…

We prove asymptotically optimal bounds on the number of edges a graph $G$ must have in order that any $r$-colouring of $E(G)$ has a colour class which contains every $D$-degenerate graph on $n$ vertices with bounded maximum degree. We also…

Let $F$ be a graph which contains an edge whose deletion reduces its chromatic number. We prove tight bounds on the number of copies of $F$ in a graph with a prescribed number of vertices and edges. Our results extend those of Simonovits,…