Related papers: Hamiltonian Berge cycles in random hypergraphs

We study Hamiltonicity in graphs obtained as the union of a deterministic $n$-vertex graph $H$ with linear degrees and a $d$-dimensional random geometric graph $G^d(n,r)$, for any $d\geq1$. We obtain an asymptotically optimal bound on the…

We prove that for integers $2 \leq \ell < k$ and a small constant $c$, if a $k$-uniform hypergraph with linear minimum codegree is randomly `perturbed' by changing non-edges to edges independently at random with probability $p \geq…

One of the most famous results in the theory of random graphs establishes that the threshold for Hamiltonicity in the Erdos-Renyi random graph G_{n,p} is around p ~ (log n + log log n) / n. Much research has been done to extend this to…

We show that for $ \eta>0 $ and sufficiently large $ n $, every 5-graph on $ n $ vertices with $\delta_{2}(H)\ge (91/216+\eta)\binom{n}{3}$ contains a Hamilton 2-cycle. This minimum 2-degree condition is asymptotically best possible.…

It has been conjectured that for any fixed $r\geq 2$ and sufficiently large $n$, there is a monochromatic Hamiltonian Berge-cycle in every $(r-1)$-coloring of the edges of $K_{n}^{r}$, the complete $r$-uniform hypergraph on $n$ vertices. In…

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…

For a graph $G$ the random $n$-lift of $G$ is obtained by replacing each of its vertices by a set of $n$ vertices, and joining a pair of sets by a random matching whenever the corresponding vertices of $G$ are adjacent. We show that…

A Berge cycle of length $\ell$ in a hypergraph is an alternating sequence of $\ell$ distinct vertices and $\ell$ distinct edges $v_1,e_1,v_2, \ldots, v_\ell, e_{\ell}$ such that $\{v_i, v_{i+1}\} \subseteq e_i$ for all $i$, with indices…

In this paper, we consider a random geometric graph (RGG)~\(G\) on~\(n\) nodes with adjacency distance~\(r_n\) just below the Hamiltonicity threshold and construct Hamiltonian cycles using additional edges called bridges. The bridges by…

In this paper we study the maximum number of hyperedges which may be in an $r$-uniform hypergraph under the restriction that no pair of vertices has more than $t$ Berge paths of length $k$ between them. When $r=t=2$, this is the even-cycle…

We show that every $3$-uniform hypergraph $H=(V,E)$ with $|V(H)|=n$ and minimum pair degree at least $(4/5+o(1))n$ contains a squared Hamiltonian cycle. This may be regarded as a first step towards a hypergraph version of the P\'osa-Seymour…

We show that the probability that a random graph $G\sim G(n,p)$ contains no Hamilton cycle is $(1+o(1))Pr(\delta (G) < 2)$ for all values of $p = p(n)$. We also prove an analogous result for perfect matchings.

Dirac proved that each $n$-vertex $2$-connected graph with minimum degree $k$ contains a cycle of length at least $\min\{2k, n\}$. We obtain analogous results for Berge cycles in hypergraphs. Recently, the authors proved an exact lower…

We study the emergence of loose Hamilton cycles in subgraphs of random hypergraphs. Our main result states that the minimum $d$-degree threshold for loose Hamiltonicity relative to the random $k$-uniform hypergraph $H_k(n,p)$ coincides with…

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…

In 1999, Jacobson and Lehel conjectured that for $k \geq 3$, every $k$-regular Hamiltonian graph has cycles of at least linearly many different lengths. This was further strengthened by Verstra\"{e}te, who asked whether the regularity can…

In this paper, we study discrepancy questions for spanning subgraphs of $k$-uniform hypergraphs. Our main result is that, for any integers $k \ge 3$ and $r \ge 2$, any $r$-colouring of the edges of a $k$-uniform $n$-vertex hypergraph $G$…

We show that every 3-uniform hypergraph with minimum vertex degree at least $0.8\binom{n-1}{2}$ contains a tight Hamiltonian cycle.

We study the existence of powers of Hamiltonian cycles in graphs with large minimum degree to which some additional edges have been added in a random manner. It follows from the theorems of Dirac and of Koml\'os, Sark\"ozy, and Szemer\'edi…

We describe an algorithm for finding Hamilton cycles in random graphs. Our model is the random graph $G=\gc$. In this model $G$ is drawn uniformly from graphs with vertex set $[n]$, $m$ edges and minimum degree at least three. We focus on…