Related papers: Hamiltonian Berge cycles in random hypergraphs

Let $G$ be a graph obtained as the union of some $n$-vertex graph $H_n$ with minimum degree $\delta(H_n)\geq\alpha n$ and a $d$-dimensional random geometric graph $G^d(n,r)$. We investigate under which conditions for $r$ the graph $G$ will…

Let $X_1,X_2,\ldots,X_n$ be chosen independently and uniformly at random from the unit $d$-dimensional cube $[0,1]^d$. Let $r$ be given and let $\cal X=\{X_1,X_2,\ldots,X_n\}$. The random geometric graph $G=G_{\cal X,r}$ has vertex set…

A tight cycle in an $r$-uniform hypergraph $\mathcal{H}$ is a sequence of $\ell\geq r+1$ vertices $x_1,\dots,x_{\ell}$ such that all $r$-tuples $\{x_{i},x_{i+1},\dots,x_{i+r-1}\}$ (with subscripts modulo $\ell$) are edges of $\mathcal{H}$.…

Ore's Theorem states that if $G$ is an $n$-vertex graph and every pair of non-adjacent vertices has degree sum at least $n$, then $G$ is Hamiltonian. A $[3]$-graph is a hypergraph in which every edge contains at most $3$ vertices. In this…

We study properties of random subcomplexes of partitions returned by (a suitable form of) the Strong Hypergraph Regularity Lemma, which we call regular slices. We argue that these subcomplexes capture many important structural properties of…

We present a tight extremal threshold for the existence of Hamilton cycles in graphs with large minimum degree and without a large ``bipartite hole`` (two disjoint sets of vertices with no edges between them). This result extends Dirac's…

We prove a conjecture of Penrose about the standard random geometric graph process, in which n vertices are placed at random on the unit square and edges are sequentially added in increasing order of lengths taken in the l_p norm. We show…

In the 1960s, Erd\H{o}s and Gallai conjectured that the edge set of every graph on n vertices can be partitioned into O(n) cycles and edges. They observed that one can easily get an O(n log n) upper bound by repeatedly removing the edges of…

Let $n\geq k\geq r+3$ and $\mathcal H$ be an $n$-vertex $r$-uniform hypergraph. We show that if $|\mathcal H|> \frac{n-1}{k-2}\binom{k-1}{r}$ then $\mathcal H$ contains a Berge cycle of length at least $k$. This bound is tight when $k-2$…

We consider the random graph $G_{n, {\bf d}}$ chosen uniformly at random from the set of all graphs with a given sparse degree sequence ${\bf d}$. We assume ${\bf d}$ has minimum degree at least 4, at most a power law tail, and place one…

We study the structure of $r$-uniform hypergraphs containing no Berge cycles of length at least $k$ for $k \leq r$, and determine that such hypergraphs have some special substructure. In particular we determine the extremal number of such…

We show that for any fixed $\alpha>0$, cherry-quasirandom 3-graphs of positive density and sufficiently large order $n$ with minimum vertex degree $\alpha \binom n2$ have a tight Hamilton cycle. This solves a conjecture of Aigner-Horev and…

We study Hamiltonicity and pancyclicity in the graph obtained as the union of a deterministic $n$-vertex graph $H$ with $\delta(H)\geq\alpha n$ and a random $d$-regular graph $G$, for $d\in\{1,2\}$. When $G$ is a random $2$-regular graph,…

Let $EG_r(n,k)$ denote the maximum number of edges in an $n$-vertex $r$-uniform hypergraph with no Berge cycles of length $k$ or longer. In the first part of this work, we have found exact values of $EG_r(n,k)$ and described the structure…

We show that, for a natural notion of quasirandomness in $k$-uniform hypergraphs, any quasirandom $k$-uniform hypergraph on $n$ vertices with constant edge density and minimum vertex degree $\Omega(n^{k-1})$ contains a loose Hamilton cycle.…

The famous Posa conjecture states that every graph of minimum degree at least 2n/3 contains the square of a Hamilton cycle. This has been proved for large n by Koml\'os, Sark\"ozy and Szemer\'edi. Here we prove that if p > n^{-1/2+\eps},…

We find Dirac-type sufficient conditions for a hypergraph $\mathcal H$ with few edges to be hamiltonian. We also show that these conditions provide that $\mathcal H$ is {\em super-pancyclic}, i.e., for each $A \subseteq V(\mathcal H)$ with…

In this note we obtain upper bounds on the number of hyperedges in 3-uniform hypergraphs not containing a Berge cycle of given odd length. We improve the bound given by F\"uredi and \"Ozkahya in 2017. The result follows from a more general…

We prove that any $3$-uniform hypergraph whose minimum vertex degree is at least $\left(\frac{5}{9} + o(1) \right)\binom{n}{2}$ admits an almost-spanning tight cycle, that is, a tight cycle leaving $o(n)$ vertices uncovered. The bound on…

We study the question of the least number of random edges that need to be added to a P\'osa-Seymour graph, that is, a graph with minimum degree exceeding $\frac k{k+1}n$, to secure the existence of the $m$-th power of a Hamiltonian cycle,…