Related papers: Hypergraphs with no tight cycles

For positive integers $r > \ell$, an $r$-uniform hypergraph is called an $\ell$-cycle if there exists a cyclic ordering of its vertices such that each of its edges consists of $r$ consecutive vertices, and such that every pair of…

It is a classical result that a random permutation of $n$ elements has, on average, about $\log n$ cycles. We generalise this fact to all directed $d$-regular graphs on $n$ vertices by showing that, on average, a random cycle-factor of such…

The Tur\'an problem asks for the largest number of edges in an $n$-vertex graph not containing a fixed forbidden subgraph $F$. We construct a new family of graphs not containing $K_{s,t}$, for $t= C^s$, with $\Omega(n^{2-1/s})$ edges…

We study the topic of "extremal" planar graphs, defining $\mathrm{ex_{_{\mathcal{P}}}}(n,H)$ to be the maximum number of edges possible in a planar graph on $n$ vertices that does not contain a given graph $H$ as a subgraph. In…

We produce an edge-coloring of the complete 3-uniform hypergraph on n vertices with $e^{O(\sqrt {log log n})}$ colors such that the edges spanned by every set of five vertices receive at least three distinct colors. This answers the first…

Given a graph $G$, a Berge copy of $G$ is a hypergraph obtained by enlarging the edges arbitrarily. Gy\H ori in 2006 showed that for $r=3$ or $r=4$, an $r$-uniform $n$-vertex Berge triangle-free hypergraph has at most $\lfloor…

We study sufficient conditions for the existence of Hamilton cycles in uniformly dense $3$-uniform hypergraphs. Problems of this type were first considered by Lenz, Mubayi, and Mycroft for loose Hamilton cycles and Aigner-Horev and Levy…

An $r$-uniform hypergraph is called an $r$-graph. A hypergraph is linear if every two edges intersect in at most one vertex. Given a linear $r$-graph $H$ and a positive integer $n$, the linear Tur\'an number $ex_L(n,H)$ is the maximum…

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…

In this note we asymptotically determine the maximum number of hyperedges possible in an $r$-uniform, connected $n$-vertex hypergraph without a Berge path of length $k$, as $n$ and $k$ tend to infinity. We show that, unlike in the graph…

For a family $\mathcal{F}$ of graphs, let $ex(n,\mathcal{F})$ denote the maximum number of edges in an $n$-vertex graph which contains none of the members of $\mathcal{F}$ as a subgraph. A longstanding problem in extremal graph theory asks…

Given $r$-uniform hypergraphs $G$ and $H$ the Tur\'an number $\rm ex(G, H)$ is the maximum number of edges in an $H$-free subgraph of $G$. We study the typical value of $\rm ex(G, H)$ when $G=G_{n,p}^{(r)}$, the Erd\H{o}s-R\'enyi random…

Given a family of $k$-hypergraphs $\mathcal{F}$, $ex(n,\mathcal{F})$ is the maximum number of edges a $k$-hypergraph can have, knowing that said hypergraph has $n$ vertices but contains no copy of any hypergraph from $\mathcal{F}$ as a…

Let $ex_r(N,g)$ be the maximum number of edges in an $r$-uni\-form hypergraph on $N$ vertices with girth at least $g$. We are interested in the asymptotic behavior of this value when $N$ is increasing but parameters $g\in\{6,8\}$ and…

The rainbow Tur\'an number $\mathrm{ex}^*(n,H)$ of a graph $H$ is the maximum possible number of edges in a properly edge-coloured $n$-vertex graph with no rainbow subgraph isomorphic to $H$. We prove that for any integer $k\geq 2$,…

We study a model of random uniform hypergraphs, where a random instance is obtained by adding random edges to a large hypergraph of a given density. We obtain a tight bound on the number of random edges required to ensure…

We investigate the occurrence of powers of tight Hamilton cycles in random hypergraphs. For every $r\ge 3$ and $k\ge 1$, we show that there exists a constant $C > 0$ such that if $p=p(n) \ge Cn^{-1/\binom{k+r-2}{r-1}}$ then asymptotically…

We explore properties of $3$-uniform hypergraphs $H$ without linear cycles. Our main results are that these hypergraphs must contain a vertex of strong degree at most two and must have independent sets of size at least ${2|V(H)|\over 5}$.

The Harary reconstruction conjecture states that any graph with more than four edges can be uniquely reconstructed from its set of maximal edge-deleted subgraphs. In 1977, M\"uller verified the conjecture for graphs with $n$ vertices and $n…

Let $\mathcal{H}$ be a $k$-uniform hypergraph. A chain in $\mathcal{H}$ is a sequence of its vertices such that every $k$ consecutive vertices form an edge. In 1999 Katona and Kierstead suggested to use chains in hypergraphs as the…