Related papers: On Weak Hamiltonicity of a Random Hypergraph

Compared to the classical binomial random (hyper)graph model, the study of random regular hypergraphs is made more challenging due to correlations between the occurrence of different edges. We develop an edge-switching technique for…

Denote by $r_g(G,\mathcal{H})$ the global resilience of a graph $G$ with respect to Hamiltonicity. That is, $r_g(G,\mathcal{H})$ is the minimal $r$ for which there exists a subgraph $H\subseteq G$ with $r$ edges, such that $G\setminus H$ is…

We present several results in extremal graph and hypergraph theory of topological nature. First, we show that if $\alpha>0$ and $\ell=\Omega(\frac{1}{\alpha}\log\frac{1}{\alpha})$ is an odd integer, then every graph $G$ with $n$ vertices…

We investigate 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. For all integers $k\geq1$, $r\geq 0$, and $\ell\geq (r+1)r$, and for any…

In 1999, Katona and Kierstead conjectured that if a $k$-uniform hypergraph $\cal H$ on $n$ vertices has minimum co-degree $\lfloor \frac{n-k+3}{2}\rfloor$, i.e., each set of $k-1$ vertices is contained in at least $\lfloor…

Dirac proved that each $n$-vertex $2$-connected graph with minimum degree at least $k$ contains a cycle of length at least $\min\{2k, n\}$. We consider a hypergraph version of this result. A Berge cycle in a hypergraph is an alternating…

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 prove packing and counting theorems for arbitrarily oriented Hamilton cycles in ${\cal D}(n,p)$ for nearly optimal $p$ (up to a $\log ^cn$ factor). In particular, we show that given $t = (1-o(1))np$ Hamilton cycles $C_1,\ldots ,C_{t}$,…

Let $\mathcal{H}$ be a given finite (possibly empty) family of connected graphs, each containing a cycle, and let $G$ be an arbitrary finite $\mathcal{H}$-free graph with minimum degree at least $k$. For $p \in [0,1]$, we form a $p$-random…

A Hamilton cycle in a digraph is a cycle that passes through all the vertices, where all the arcs are oriented in the same direction. The problem of finding Hamilton cycles in directed graphs is well studied and is known to be hard. One of…

Given an $n$ vertex graph whose edges have colored from one of $r$ colors $C=\{c_1,c_2,\ldots,c_r\}$, we define the Hamilton cycle color profile $hcp(G)$ to be the set of vectors $(m_1,m_2,\ldots,m_r)\in [0,n]^r$ such that there exists a…

A seminal result by Koml\'os, Sark\"ozy, and Szemer\'edi states that if a graph $G$ with $n$ vertices has minimum degree at least $kn/(k + 1)$, for some $k \in \mathbb{N}$ and $n$ sufficiently large, then it contains the $k$-th power of a…

A graph is Hamiltonian if it contains a cycle which passes through every vertex of the graph exactly once. A classical theorem of Dirac from 1952 asserts that every graph on $n$ vertices with minimum degree at least $n/2$ is Hamiltonian. We…

A $k$-uniform hypergraph $H = (V, E)$ is called $\ell$-orientable, if there is an assignment of each edge $e\in E$ to one of its vertices $v\in e$ such that no vertex is assigned more than $\ell$ edges. Let $H_{n,m,k}$ be a hypergraph,…

The (k,d)-hypersimplex is a (d-1)-dimensional polytope whose vertices are the (0,1)-vectors that sum to k. When k=1, we get a simplex whose graph is the complete graph with d vertices. Here we show how many of the well known graph…

We consider homological edge percolation on a sequence $(\mathcal{G}_t)_t$ of finite graphs covered by an infinite (quasi)transitive graph $\mathcal{H}$, and weakly convergent to $\mathcal{H}$. Namely, we use the covering maps to classify…

It is known that the complete graph $K_n$ contains a pancyclic subgraph with $n+(1+o(1))\cdot \log _2 n$ edges, and that there is no pancyclic graph on $n$ vertices with fewer than $n+\log _2 (n-1) -1$ edges. We show that, with high…

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…

A cornerstone of extremal graph theory due to Erd\H{o}s and Stone states that the edge density which guarantees a fixed graph $F$ as subgraph also asymptotically guarantees a blow-up of $F$ as subgraph. It is natural to ask whether this…

In this paper, we study rare events in spherical and Gaussian random geometric graphs in high dimensions. In these models, the vertices correspond to points sampled uniformly at random on the $d$ dimensional unit sphere or correspond to $d$…