Related papers: The probability of selecting $k$ edge-disjoint Ham…

We say that a $k$-uniform hypergraph $C$ is a Hamilton cycle of type $\ell$, for some $1\le \ell \le k$, if there exists a cyclic ordering of the vertices of $C$ such that every edge consists of $k$ consecutive vertices and for every pair…

A subset $C$ of edges in a $k$-uniform hypergraph $H$ is a \emph{loose Hamilton cycle} if $C$ covers all the vertices of $H$ and there exists a cyclic ordering of these vertices such that the edges in $C$ are segments of that order and such…

We study the existence of directed Hamilton cycles in random digraphs with $m$ edges where we condition on minimum in- and out-degree $\d \ge k+1$, where $k \ge 1$. Denote such a random graph by $D_{n,m}^{(\delta\geq k+1)}$. Let $m=cn$ and…

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…

We show that provided $\log^{50} n/n \leq p \leq 1 - n^{-1/4}\log^9 n$ we can with high probability find a collection of $\lfloor \delta(G)/2 \rfloor$ edge-disjoint Hamilton cycles in $G \sim G_{n, p}$, plus an additional edge-disjoint…

In this paper we consider the problem of finding ``as many edge-disjoint Hamilton cycles as possible'' in the binomial random digraph $D_{n,p}$. We show that a typical $D_{n,p}$ contains precisely the minimum between the minimum out- and…

In 1963, Anton Kotzig famously conjectured that $K_{n}$, the complete graph of order $n$, where $n$ is even, can be decomposed into $n-1$ perfect matchings such that every pair of these matchings forms a Hamilton cycle. The problem is still…

Given a symmetric $n\times n$ matrix $P$ with $0 \le P(u, v)\le 1$, we define a random graph $G_{n, P}$ on $[n]$ by independently including any edge $\{u, v\}$ with probability $P(u, v)$. For $k\ge 1$ let $\mathcal{A}_k$ be the property of…

We propose the following conjecture extending Dirac's theorem: if $G$ is a graph with $n\ge 3$ vertices and minimum degree $\delta(G)\ge n/2$, then in every orientation of $G$ there is a Hamilton cycle with at least $\delta(G)$ edges…

An {\em $\ell$-offset Hamilton cycle} $C$ in a $k$-uniform hypergraph $H$ on~$n$ vertices is a collection of edges of $H$ such that for some cyclic order of $[n]$ every pair of consecutive edges $E_{i-1},E_i$ in $C$ (in the natural ordering…

In the random hypergraph $H_{n,p;k}$ each possible $k$-tuple appears independently with probability $p$. A loose Hamilton cycle is a cycle in which every pair of adjacent edges intersects in a single vertex. We prove that if $p n^{k-1}/\log…

We introduce a new setting of algorithmic problems in random graphs, studying the minimum number of queries one needs to ask about the adjacency between pairs of vertices of ${\mathcal G}(n,p)$ in order to typically find a subgraph…

We study Hamiltonicity in random subgraphs of the hypercube $\mathcal{Q}^n$. Our first main theorem is an optimal hitting time result. Consider the random process which includes the edges of $\mathcal{Q}^n$ according to a uniformly chosen…

We prove that a random graph $G(n,p)$, with $p$ above the Hamiltonicity threshold, is typically such that for any $r$-colouring of its edges there exists a Hamilton cycle with at least $(2/(r+ 1)-o(1))n$ edges of the same colour. This…

We study the number of edge-disjoint Hamilton cycles one can guarantee in a sufficiently large graph G on n vertices with minimum degree d = (1/2+a)n. For any constant a > 0, we give an optimal answer in the following sense: let…

We say that a $k$-uniform hypergraph $C$ is a Hamilton cycle of type $\ell$, for some $1\le \ell \le k$, if there exists a cyclic ordering of the vertices of $C$ such that every edge consists of $k$ consecutive vertices and for every pair…

We show how to adjust a very nice coupling argument due to McDiarmid in order to prove/reprove in a novel way results concerning Hamilton cycles in various models of random graph and hypergraphs. In particular, we firstly show that for…

In this paper we give an approximate answer to a question of Nash-Williams from 1970: we show that for every \alpha > 0, every sufficiently large graph on n vertices with minimum degree at least (1/2 + \alpha)n contains at least n/8…

We show that if pn >> log n, the binomial random graph G_{n,p} has an approximate Hamilton decomposition. More precisely, we show that in this range G_{n,p} contains a set of edge-disjoint Hamilton cycles covering almost all of its edges.…

For $0\leq \ell <k$, a Hamiltonian $\ell$-cycle in a $k$-uniform hypergraph $H$ is a cyclic ordering of the vertices of $H$ in which the edges are segments of length $k$ and every two consecutive edges overlap in exactly $\ell$ vertices. We…