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In an r-uniform hypergraph on n vertices, a tight Hamilton cycle consists of n edges such that there exists a cyclic ordering of the vertices where the edges correspond to consecutive segments of r vertices. We provide a first deterministic polynomial-time algorithm, which finds a.a.s. tight Hamilton cycles in random r-uniform hypergraphs with edge probability at least C log3n/n.
Our result partially answers a question of Dudek and Frieze, who proved that tight Hamilton cycles exist already for p = ω(1/n) for r = 3 and p = (e + o(1))/n for
$r \ge 4$
using a second moment argument. Moreover our algorithm is superior to previous results of Allen, Böttcher, Kohayakawa and Person, and Nenadov and Škorić, in various ways: the algorithm of Allen et al. is a randomized polynomial-time algorithm working for edge probabilities
$p \ge {n^{ - 1 + \varepsilon}}$
, while the algorithm of Nenadov and Škorić is a randomized quasipolynomial-time algorithm working for edge probabilities
$p \ge C\mathop {\log }\nolimits^8 n/n$
.
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 sequentially added to an initially empty r-graph, then as soon as the minimum degree is at least 2, the hypergraph with high probability 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 with high probability has such a cycle. We obtain similar results for weak Berge cycles as well, thus resolving a conjecture of Poole.
In this paper we study first passage percolation on a random graph model, the configuration model. We first introduce the notions of weighted diameter, which is the maximum of the weighted lengths of all optimal paths between any two vertices in the graph, and the flooding time, which represents the time (weighted length) needed to reach all the vertices in the graph starting from a uniformly chosen vertex. Our result consists in describing the asymptotic behavior of the diameter and the flooding time, as the number of vertices n tends to infinity, in the case where the weight distribution G has an exponential tail behavior, and proving that this category of distributions is the largest possible for which the asymptotic behavior holds.
We prove a ‘resilience’ version of Dirac’s theorem in the setting of random regular graphs. More precisely, we show that whenever d is sufficiently large compared to
$\epsilon > 0$
, a.a.s. the following holds. Let
$G'$
be any subgraph of the random n-vertex d-regular graph
$G_{n,d}$
with minimum degree at least
$$(1/2 + \epsilon )d$$
. Then
$G'$
is Hamiltonian.
This proves a conjecture of Ben-Shimon, Krivelevich and Sudakov. Our result is best possible: firstly the condition that d is large cannot be omitted, and secondly the minimum degree bound cannot be improved.
Given a fixed graph H, a real number p ∈ (0, 1) and an infinite Erdös–Rényi graph G ∼ G(∞, p), how many adjacency queries do we have to make to find a copy of H inside G with probability at least 1/2? Determining this number f(H, p) is a variant of the subgraph query problem introduced by Ferber, Krivelevich, Sudakov and Vieira. For every graph H, we improve the trivial upper bound of f(H, p) = O(p−d), where d is the degeneracy of H, by exhibiting an algorithm that finds a copy of H in time O(p−d) as p goes to 0. Furthermore, we prove that there are 2-degenerate graphs which require p−2+o(1) queries, showing for the first time that there exist graphs H for which f(H, p) does not grow like a constant power of p−1 as p goes to 0. Finally, we answer a question of Feige, Gamarnik, Neeman, Rácz and Tetali by showing that for any δ < 2, there exists α < 2 such that one cannot find a clique of order α log2n in G(n, 1/2) in nδ queries.
The triangle packing number v(G) of a graph G is the maximum size of a set of edge-disjoint triangles in G. Tuza conjectured that in any graph G there exists a set of at most 2v(G) edges intersecting every triangle in G. We show that Tuza’s conjecture holds in the random graph G = G(n, m), when m ⩽ 0.2403n3/2 or m ⩾ 2.1243n3/2. This is done by analysing a greedy algorithm for finding large triangle packings in random graphs.
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 graph is
${(1/2-\varepsilon)}$
-resiliently Hamiltonian. Furthermore, for each
$\varepsilon > 0$
, we show that, almost surely, each directed graph
$D_M$
in the sequence is not
$(1/2+\varepsilon)$
-resiliently Hamiltonian.
This improves a result of Ferber, Nenadov, Noever, Peter and Škorić who showed, for each
$\varepsilon > 0$
, that the binomial random directed graph
$D(n,p)$
is almost surely
$(1/2-\varepsilon)$
-resiliently Hamiltonian if
$p=\omega(\log^8n/n)$
.
For fixed graphs F1,…,Fr, we prove an upper bound on the threshold function for the property that G(n, p) → (F1,…,Fr). This establishes the 1-statement of a conjecture of Kohayakawa and Kreuter.
If we pick n random points uniformly in
$[0,1]^d$
and connect each point to its
$c_d \log{n}$
nearest neighbors, where
$d\ge 2$
is the dimension and
$c_d$
is a constant depending on the dimension, then it is well known that the graph is connected with high probability. We prove that it suffices to connect every point to
$ c_{d,1} \log{\log{n}}$
points chosen randomly among its
$ c_{d,2} \log{n}$
nearest neighbors to ensure a giant component of size
$n - o(n)$
with high probability. This construction yields a much sparser random graph with
$\sim n \log\log{n}$
instead of
$\sim n \log{n}$
edges that has comparable connectivity properties. This result has non-trivial implications for problems in data science where an affinity matrix is constructed: instead of connecting each point to its k nearest neighbors, one can often pick
$k'\ll k$
random points out of the k nearest neighbors and only connect to those without sacrificing quality of results. This approach can simplify and accelerate computation; we illustrate this with experimental results in spectral clustering of large-scale datasets.
A uniform recursive tree on n vertices is a random tree where each possible
$(n-1)!$
labelled recursive rooted tree is selected with equal probability. We introduce and study weighted trees, a non-uniform recursive tree model departing from the recently introduced Hoppe trees. This class generalizes both uniform recursive trees and Hoppe trees, providing diversity among the nodes and making the model more flexible for applications. We analyse the number of leaves, the height, the depth, the number of branches, and the size of the largest branch in these weighted trees.
Let V be an n-set, and let X be a random variable taking values in the power-set of V. Suppose we are given a sequence of random coupons
$X_1, X_2, \ldots $
, where the
$X_i$
are independent random variables with distribution given by X. The covering time T is the smallest integer
$t\geq 0$
such that
$\bigcup_{i=1}^t X_i=V$
. The distribution of T is important in many applications in combinatorial probability, and has been extensively studied. However the literature has focused almost exclusively on the case where X is assumed to be symmetric and/or uniform in some way.
In this paper we study the covering time for much more general random variables X; we give general criteria for T being sharply concentrated around its mean, precise tools to estimate that mean, as well as examples where T fails to be concentrated and when structural properties in the distribution of X allow for a very different behaviour of T relative to the symmetric/uniform case.
We study random unlabelled k-trees by combining the colouring approach by Gainer-Dewar and Gessel (2014) with the cycle-pointing method by Bodirsky, Fusy, Kang and Vigerske (2011). Our main applications are Gromov–Hausdorff–Prokhorov and Benjamini–Schramm limits that describe their asymptotic geometric shape on a global and local scale as the number of (k + 1)-cliques tends to infinity.
The directed preferential attachment model is revisited. A new exact characterization of the limiting in- and out-degree distribution is given by two independent pure birth processes that are observed at a common exponentially distributed time T (thus creating dependence between in- and out-degree). The characterization gives an explicit form for the joint degree distribution, and this confirms previously derived tail probabilities for the two marginal degree distributions. The new characterization is also used to obtain an explicit expression for tail probabilities in which both degrees are large. A new generalized directed preferential attachment model is then defined and analyzed using similar methods. The two extensions, motivated by empirical evidence, are to allow double-directed (i.e. undirected) edges in the network, and to allow the probability of connecting an ingoing (outgoing) edge to a specified node to also depend on the out-degree (in-degree) of that node.
We consider a simple preferential attachment graph process, which begins with a finite graph and in which a new (t + 1)st vertex is added at each subsequent time step t that is connected to each previous vertex u ≤ t with probability du(t)/t, where du(t) is the degree of u at time t. We analyse the graph obtained as the infinite limit of this process, and we show that, as long as the initial finite graph is neither edgeless nor complete, with probability 1 the outcome will be a copy of the Rado graph augmented with a finite number of either isolated or universal vertices.
We give an example of a long range Bernoulli percolation process on a group non-quasi-isometric with ℤ, in which clusters are almost surely finite for all values of the parameter. This random graph admits diverse equivalent definitions, and we study their ramifications. We also study its expected size and point out certain phase transitions.
Random constraint satisfaction problems play an important role in computer science and combinatorics. For example, they provide challenging benchmark examples for algorithms, and they have been harnessed in probabilistic constructions of combinatorial structures with peculiar features. In an important contribution (Krzakala et al. 2007, Proc. Nat. Acad. Sci.), physicists made several predictions on the precise location and nature of phase transitions in random constraint satisfaction problems. Specifically, they predicted that their satisfiability thresholds are quite generally preceded by several other thresholds that have a substantial impact both combinatorially and computationally. These include the condensation phase transition, where long-range correlations between variables emerge, and the reconstruction threshold. In this paper we prove these physics predictions for a broad class of random constraint satisfaction problems. Additionally, we obtain contiguity results that have implications for Bayesian inference tasks, a subject that has received a great deal of interest recently (e.g. Banks et al. 2016, Proc. 29th COLT).
An equitable colouring of a graph G is a vertex colouring where no two adjacent vertices are coloured the same and, additionally, the colour class sizes differ by at most 1. The equitable chromatic number χ=(G) is the minimum number of colours required for this. We study the equitable chromatic number of the dense random graph
${\mathcal{G}(n,m)}$
where
$m = \left\lfloor {p\left( \matrix{
n \cr
2 \cr} \right)} \right\rfloor $
and 0 < p < 0.86 is constant. It is a well-known question of Bollobás [3] whether for p = 1/2 there is a function f(n) → ∞ such that, for any sequence of intervals of length f(n), the normal chromatic number of
${\mathcal{G}(n,m)}$
lies outside the intervals with probability at least 1/2 if n is large enough. Bollobás proposes that this is likely to hold for f(n) = log n. We show that for the equitable chromatic number, the answer to the analogous question is negative. In fact, there is a subsequence
${({n_j})_j}_{ \in {\mathbb {N}}}$
of the integers where
$\chi_=({\mathcal{G}(n_j,m_j)})$
is concentrated on exactly one explicitly known value. This constitutes surprisingly narrow concentration since in this range the equitable chromatic number, like the normal chromatic number, is rather large in absolute value, namely asymptotically equal to n/(2logbn) where b = 1/(1 − p).
We prove that, with high probability, in every 2-edge-colouring of the random tournament on n vertices there is a monochromatic copy of every oriented tree of order
$O(n{\rm{/}}\sqrt {{\rm{log}} \ n} )$
. This generalizes a result of the first, third and fourth authors, who proved the same statement for paths, and is tight up to a constant factor.
Given a graph G and a bijection f : E(G) → {1, 2,…,e(G)}, we say that a trail/path in G is f-increasing if the labels of consecutive edges of this trail/path form an increasing sequence. More than 40 years ago Chvátal and Komlós raised the question of providing worst-case estimates of the length of the longest increasing trail/path over all edge orderings of Kn. The case of a trail was resolved by Graham and Kleitman, who proved that the answer is n-1, and the case of a path is still wide open. Recently Lavrov and Loh proposed studying the average-case version of this problem, in which the edge ordering is chosen uniformly at random. They conjectured (and Martinsson later proved) that such an ordering with high probability (w.h.p.) contains an increasing Hamilton path.
In this paper we consider the random graph G = Gn,p with an edge ordering chosen uniformly at random. In this setting we determine w.h.p. the asymptotics of the number of edges in the longest increasing trail. In particular we prove an average-case version of the result of Graham and Kleitman, showing that the random edge ordering of Kn has w.h.p. an increasing trail of length (1-o(1))en, and that this is tight. We also obtain an asymptotically tight result for the length of the longest increasing path for random Erdős-Renyi graphs with p = o(1).
In this work, we use rigorous probabilistic methods to study the asymptotic degree distribution, clustering coefficient, and diameter of geographical attachment networks. As a type of small-world network model, these networks were first proposed in the physical literature, where they were analyzed only with heuristic arguments and computational simulations.