Let $X_1,..., X_n$ be independent, uniformly random points from $[0,1]^2$. We prove that if we add edges between these points one by one by order of increasing edge length then, with probability tending to 1 as the number of points $n$ tends to $\infty$, the resulting graph gets its first Hamilton cycle at exactly the same time it loses its last vertex of degree less than two. This answers an open question of Penrose and provides an analogue for the random geometric graph of a celebrated result of Ajtai, Komlós and Szemerédi and independently of Bollobás on the usual random graph. We are also able to deduce very precise information on the limiting probability that the random geometric graph is Hamiltonian analogous to a result of Komlós and Szemer{é}di on the usual random graph. The proof generalizes to uniform random points on the $d$-dimensional hypercube where the edge-lengths are measured using the $l_p$-norm for some $1<p\leq\infty$. The proof can also be adapted to show that, with probability tending to 1 as the number of points $n$ tends to $\infty$, there are cycles of all lengths between 3 and $n$ at the moment the graph loses its last vertex of degree less than two.
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 minimum $r$ for which a.a.s. $H\cup G^d(n,r)$ is Hamiltonian. Our proof provides a linear time algorithm to find a Hamilton cycle in such graphs.
We study the Hamiltonicity of the following model of a random graph. Suppose that we partition [n] into V_1,V_2,...,V_k and add edge {x,y} to our graph with probability p if there exists i such that x,y\in V_i. Otherwise, we add the edge with probbability q. We denote this model by G(n, p,q) and give tight results for Hamiltonicity, including a critical window analysis, under various conditions.
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 more condition on its tail. For $k\ge 2$ define $β_k(G) = \max e(A, B) + k(|A|-|B|) - d(A)$, with the maximum taken over disjoint vertex sets $A, B$. It is shown that the problem of determining if $G_{n, {\bf d}}$ contains a Hamilton cycle reduces to calculating $β_2(G_{n, {\bf d}})$. If $k\ge 2$ and $δ\ge k+2$, the problem of determining if $G_{n, {\bf d}}$ c…
We show that the probability that a random graph $G\sim G(n,p)$ contains no Hamilton cycle is $(1+o(1))Pr(δ(G) < 2)$ for all values of $p = p(n)$. We also prove an analogous result for perfect matchings.
In this paper we study the toughness of Random Apollonian Networks (RANs), a random graph model which generates planar graphs with power-law properties. We consider their important characteristics: every RAN is a uniquely representable chordal graph and a planar $3$-tree and as so, known results about these classes can be particularized. We establish a partition of the class in eight nontrivial subclasses and for each one of these subclasses we provide bounds for the toughness of their elements. We also study the hamiltonicity of the elements of these subclasses.
We study Hamiltonicity and pancyclicity in the graph obtained as the union of a deterministic $n$-vertex graph $H$ with $δ(H)\geqαn$ and a random $d$-regular graph $G$, for $d\in\{1,2\}$. When $G$ is a random $2$-regular graph, we prove that a.a.s. $H\cup G$ is pancyclic for all $α\in(0,1]$, and also extend our result to a range of sublinear degrees. When $G$ is a random $1$-regular graph, we prove that a.a.s. $H\cup G$ is pancyclic for all $α\in(\sqrt{2}-1,1]$, and this result is best possible. Furthermore, we show that this bound on $δ(H)$ is only needed when $H$ is `far' from containing a p…
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 random ordering. Then, with high probability, as soon as the graph produced by this process has minimum degree $2k$, it contains $k$ edge-disjoint Hamilton cycles, for any fixed $k\in\mathbb{N}$. Secondly, we obtain a perturbation result: if $H\subseteq\mathcal{Q}^n$ satisfies $δ(H)\geqαn$ with $α>0$ fixed and we consider a random binomial subgraph $\mathcal{…
We prove that random hypergraphs are asymptotically almost surely resiliently Hamiltonian. Specifically, for any $γ>0$ and $k\ge3$, we show that asymptotically almost surely, every subgraph of the binomial random $k$-uniform hypergraph $G^{(k)}\big(n,n^{γ-1}\big)$ in which all $(k-1)$-sets are contained in at least $\big(\tfrac12+2γ\big)pn$ edges has a tight Hamilton cycle. This is a cyclic ordering of the $n$ vertices such that each consecutive $k$ vertices forms an edge.
Let $\{G_M\}_{M\geq 0}$ be the random graph process, where $G_0$ is the empty graph on $n$ vertices and subsequent graphs in the sequence are obtained by adding a new edge uniformly at random. For each $\varepsilon>0$, we show that, almost surely, any graph $G_M$ with minimum degree at least 2 is not only Hamiltonian (as shown by Bollobás), but remains Hamiltonian despite the removal of any set of edges, as long as at most $(1/2-\varepsilon)$ of the edges incident to each vertex are removed. We say that such a graph is $(1/2-\varepsilon)$-resiliently Hamiltonian. Furthermore, for each $ε>0$, w…
Let $\mathcal{G}(n,r,s)$ denote a uniformly random $r$-regular $s$-uniform hypergraph on $n$ vertices, where $s$ is a fixed constant and $r=r(n)$ may grow with $n$. An $\ell$-overlapping Hamilton cycle is a Hamilton cycle in which successive edges overlap in precisely $\ell$ vertices, and 1-overlapping Hamilton cycles are called loose Hamilton cycles. When $r,s\geq 3$ are fixed integers, we establish a threshold result for the property of containing a loose Hamilton cycle. This partially verifies a conjecture of Dudek, Frieze, Rucinski and Sileikis (2015). In this setting, we also find the asy…
Let $\{D_M\}_{M\geq 0}$ be the $n$-vertex random directed graph process, where $D_0$ is the empty directed graph on $n$ vertices, and subsequent directed graphs in the sequence are obtained by the addition of a new directed edge uniformly at random. For each $\varepsilon>0$, we show that, almost surely, any directed graph $D_M$ with minimum in- and out-degree at least 1 is not only Hamiltonian (as shown by Frieze), but remains Hamiltonian when edges are removed, as long as at most $(1/2-\varepsilon)$ of both the in- and out-edges incident to each vertex are removed. We say such a directed grap…
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 estimate is asymptotically optimal.
Let $ε>0$. We consider the problem of constructing a Hamiltonian graph with $(1+ε)n$ edges in the following controlled random graph process. Starting with the empty graph on $[n]$, at each round a set of $K=K(n)$ edges is presented, chosen uniformly at random from the missing ones (or from the ones that have not been presented yet), and we are asked to choose at most one of them and add it to the current graph. We show that in this process one can build a Hamiltonian graph with at most $(1+ε)n$ edges in $(1+ε)(1+(\log n)/2K) n$ rounds w.h.p. The case $K=1$ implies that w.h.p. one can build a H…
We introduce a natural generalization of the Erdős-Rényi random graph model in which random instances of a fixed motif are added independently. The binomial random motif graph $G(H,n,p)$ is the random (multi)graph obtained by adding an instance of a fixed graph $H$ on each of the copies of $H$ in the complete graph on $n$ vertices, independently with probability $p$. We establish that every monotone property has a threshold in this model, and determine the thresholds for connectivity, Hamiltonicity, the existence of a perfect matching, and subgraph appearance. Moreover, in the first three case…
In this note, we study the emergence of Hamiltonian Berge cycles in random $r$-uniform hypergraphs. For $r\geq 3$, we prove an optimal stopping-time result that if edges are sequently added to an initially empty $r$-graph, then as soon as the minimum degree is at least 2, the hypergraph almost surely has such a cycle. In particular, this determines the threshold probability for Berge Hamiltonicity of the Erdős--Rényi random $r$-graph, and we also show that the $2$-out random $r$-graph almost surely has such a cycle. We obtain similar results for \textit{weak Berge} cycles as well, thus resolvi…
Let $G_n$ be a random geometric graph, and then for $q,p \in [0,1)$ we construct a "$(q,p)$-perturbed noisy random geometric graph" $G_n^{q,p}$ where each existing edge in $G_n$ is removed with probability $q$, while and each non-existent edge in $G_n$ is inserted with probability $p$. We give asymptotically tight bounds on the clique number $ω\left(G_n^{q,p}\right)$ for several regimes of parameter.
In this paper, we study orthogonal colourings of random geometric graphs. Two colourings of a graph are orthogonal if they have the property that when two vertices receive the same colour in one colouring, then those vertices receive distinct colours in the other colouring. A random geometric graph $RG(n,r)$ is a graph constructed by randomly placing $n$ vertices in the unit square and connecting two vertices with an edge if and only if their distance is less than the threshold $r$. We show first that random geometric graphs with $r>n^{-α}$, where $0\leq α\leq\frac{1}{4}$, have an orthogonal c…
We propose the following conjecture extending Dirac's theorem: if $G$ is a graph with $n\ge 3$ vertices and minimum degree $δ(G)\ge n/2$, then in every orientation of $G$ there is a Hamilton cycle with at least $δ(G)$ edges oriented in the same direction. We prove an approximate version of this conjecture, showing that minimum degree $n/2 + O(k)$ guarantees a Hamilton cycle with at least $(n+k)/2$ edges oriented in the same direction. We also study the analogous problem for random graphs, showing that if the edge probability $p = p(n)$ is above the Hamiltonicity threshold, then, with high prob…
Given a dense countable set in a metric space, the infinite random geometric graph is the random graph with the given vertex set and where any two points at distance less than 1 are connected, independently, with some fixed probability. It has been observed by Bonato and Janssen that in some, but not all, such settings, the resulting graph does not depend on the random choices, in the sense that it is almost surely isomorphic to a fixed graph. While this notion makes sense in the general context of metric spaces, previous work has been restricted to sets in Banach spaces. We study the case whe…
