Related papers: Hamiltonicity of Sparse Pseudorandom Graphs

We consider the geometric random (GR) graph on the $d-$dimensional torus with the $L_\sigma$ distance measure ($1 \leq \sigma \leq \infty$). Our main result is an exact characterization of the probability that a particular labeled cycle…

A classical result of Dirac says that every $n$-vertex graph with minimum degree at least $\frac{n}{2}$ contains a Hamilton cycle. A `discrepancy' version of Dirac's theorem was shown by Balogh--Csaba--Jing--Pluh\'ar,…

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…

A tuple $(G_1,\dots,G_n)$ of graphs on the same vertex set of size $n$ is said to be Hamilton-universal if for every map $\chi: [n]\to[n]$ there exists a Hamilton cycle whose $i$-th edge comes from $G_{\chi(i)}$. Bowtell, Morris, Pehova and…

We prove that, in the Gilbert model for a random geometric graph, almost every graph becomes Hamiltonian exactly when it first becomes 2-connected. This answers a question of Penrose. We also show that in the k-nearest neighbor model, there…

A tight Hamilton cycle in a $k$-uniform hypergraph ($k$-graph) $G$ is a cyclic ordering of the vertices of $G$ such that every set of $k$ consecutive vertices in the ordering forms an edge. R\"{o}dl, Ruci\'{n}ski, and Szemer\'{e}di proved…

Let H be a 3-uniform hypergraph with N vertices. A tight Hamilton cycle C \subset H is a collection of N edges for which there is an ordering of the vertices v_1, ..., v_N such that every triple of consecutive vertices {v_i, v_{i+1},…

The cycle space of a graph $G$, denoted $C(G)$, is a vector space over ${\mathbb F}_2$, spanned by all incidence vectors of edge-sets of cycles of $G$. If $G$ has $n$ vertices, then $C_n(G)$ denotes the subspace of $C(G)$, spanned by the…

We prove that for every $\varepsilon > 0$ there exists $n_0=n_0(\varepsilon)$ such that every regular oriented graph on $n > n_0$ vertices and degree at least $(1/4 + \varepsilon)n$ has a Hamilton cycle. This establishes an approximate…

Let $H_r(n,p)$ denote the maximum number of Hamiltonian cycles in an $n$-vertex $r$-graph with density $p \in (0,1)$. The expected number of Hamiltonian cycles in the random $r$-graph model $G_r(n,p)$ is $E(n,p)=p^n(n-1)!/2$ and in the…

We say that a $d$-regular graph is a $\gamma$-expander if for every not too large set of vertices $S$, there are at least $\gamma d |S|$ edges leaving $S$, and we say that a graph $G$ is $\gamma$-far from bipartite if at least $\gamma e(G)$…

Given a family of graphs $G_1,\dots,G_{n}$ on the same vertex set $[n]$, a rainbow Hamilton cycle is a Hamilton cycle on $[n]$ such that each $G_c$ contributes exactly one edge. We prove that if $G_1,\dots,G_{n}$ are independent samples 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 $C_1,\ldots ,C_{t}$,…

The notion of robust expansion has played a central role in the solution of several conjectures involving the packing of Hamilton cycles in graphs and directed graphs. These and other results usually rely on the fact that every robustly…

Let D be the circulant digraph with n vertices and connection set {2,3,c}. (Assume D is loopless and has outdegree 3.) Work of S.C.Locke and D.Witte implies that if n is a multiple of 6, c is either (n/2) + 2 or (n/2) + 3, and c is even,…

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…

Let $G$ be a graph on $n\geq 3$ vertices, claw the bipartite graph $K_{1,3}$, and $Z_i$ the graph obtained from a triangle by attaching a path of length $i$ to its one vertex. $G$ is called 1-heavy if at least one end vertex of each induced…

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…

We present a polynomial space Monte Carlo algorithm that given a directed graph on $n$ vertices and average outdegree $\delta$, detects if the graph has a Hamiltonian cycle in $2^{n-\Omega(\frac{n}{\delta})}$ time. This asymptotic scaling…

Let $k\geq 2$. We show that, for a sufficiently small $\varepsilon>0$, any sufficiently large $n$-vertex Hamiltonian graph of minimum degree at least $n^{1-\varepsilon}$ contains a $2$-factor consisting of exactly $k$ cycles. This is the…