Counting Hamilton cycles in sparse random directed graphs

Let$\{D_M\}_{M\geq 0}$be then-vertex random directed graph process, where$D_0$is the empty directed graph onnvertices, 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$p=\omega(\log^8n/n)$.

We study the resilience of random and pseudorandom directed graphs with respect to the property of having long directed cycles. For every 08γ81/2 we find a constantc=c(γ) such that the following holds. LetG= (V, E) be a (pseudo)random directed graph onnvertices and with at least a linear number of edges, and letG′ be a subgraph ofGwith (1/2 + γ)|E| edges. ThenG′ contains a directed cycle of length at least (c−o(1))n. Moreover, there is a subgraphG′′ofGwith (1/2 + γ −o(1))|E| edges that does not contain a cycle of length at leastcn. © 2011 Wiley Periodicals, Inc. J Graph Theory 70: 284–296, 2012

We prove that the number of Hamilton cycles in the random graph $G(n,p)$ is $n!p^n(1+o(1))^n$ asymptotically almost surely (a.a.s.), provided that $p\geq \frac{\ln n+\ln\ln n+\omega(1)}{n}$. Furthermore, we prove the hitting time version of this statement, showing that in the random graph process, the edge that creates a graph of minimum degree $2$ creates $(\frac{\ln n}{e})^n(1+o(1))^n$ Hamilton cycles a.a.s.

We present a linear stability analysis of stationary states (or fixed points) in large dynamical systems defined on random directed graphs with a prescribed distribution of indegrees and outdegrees. We obtain two remarkable results for such dynamical systems: First, infinitely large systems on directed graphs can be stable even when the degree distribution has unbounded support; this result is surprising since their counterparts on nondirected graphs are unstable when system size is large enough. Second, we show that the phase transition between the stable and unstable phase is universal in the sense that it depends only on a few parameters, such as, the mean degree and a degree correlation coefficient. In addition, in the unstable regime we characterize the nature of the destabilizing mode, which also exhibits universal features. These results follow from an exact theory for the leading eigenvalue of infinitely large graphs that are locally tree-like and oriented, as well as, for the right and left eigenvectors associated with the leading eigenvalue. We corroborate analytical results for infinitely large graphs with numerical experiments on random graphs of finite size. We discuss how the presented theory can be extended to graphs with diagonal disorder and to graphs that contain nondirected links. Finally, we discuss the influence of small cycles and how they can destabilize large dynamical systems when they induce strong enough feedback loops.

Consider a directed analogue of the random graph process on $n$ vertices, where the $n(n-1)$ edges are ordered uniformly at random and revealed one at a time. It is known that with high probability (w.h.p.) the first digraph in this process with both in-degree and out-degree $\geq q$ has a $q$-edge-coloring with a Hamilton cycle in each color. We show that this coloring can be constructed online, where each edge must be irrevocably colored as soon as it appears. In a similar fashion, for the undirected random graph process, we present an online $n$-edge-coloring algorithm which yields w.h.p. $q$ disjoint rainbow Hamilton cycles in the first graph containing $q$ disjoint Hamilton cycles.

A random graph with (1+e)n/2 edges contains a path of lengthcn. A random directed graph with (1+e)n edges contains a directed path of lengthcn. This settles a conjecture of Erdos.

We develop techniques to prove lower bounds for the BCAST(log n) Broadcast Congested Clique model (a distributed message passing model where in each round, each processor can broadcast an O(log n)-sized message to all other processors). Our techniques are built to prove bounds for natural input distributions. So far, all lower bounds for problems in the model relied on constructing specifically tailored graph families for the specific problem at hand, resulting in lower bounds for artificially constructed inputs, instead of natural input distributions.One of our results is a lower bound for the directed planted clique problem. In this problem, an input graph is either a random directed graph (each directed edge is included with probability 1/2), or a random graph with a planted clique of size k. That is, k randomly chosen vertices have all of the edges between them included, and all other edges in the graph appear with probability 1/2. The goal is to determine whether a clique exists. We show that when k = n(1/4 - e), this problem requires a number of rounds polynomial in n.Additionally, we construct a pseudo-random generator which fools the Broadcast Congested Clique. This allows us to show that every k round randomized algorithm in which each processor uses up to n random bits can be efficiently transformed into an O(k)-round randomized algorithm in which each processor uses only up to O(k log n) random bits, while maintaining a high success probability. The pseudo-random generator is simple to describe, computationally very cheap, and its seed size is optimal up to constant factors. However, the analysis is quite involved, and is based on the new technique for proving lower bounds in the model.The technique also allows us to prove the first average case lower bound for the Broadcast Congested Clique, as well as an average-case time hierarchy. We hope our technique will lead to more lower bounds for problems such as triangle counting, APSP, MST, diameter, and more, for natural input distributions.

ight about an estimated weighted mean orbit element formation barycenter is investigated. Consensus over random directed graphs in the presence of time delays is reviewed and applied to the problem. On-orbit spacecraft formations are considered random directed graphs with time delays and agreement over formation parameters is shown. Consensus of formation state estimates and formation barycenter using a distributed formation sensor network is proven. A simulation is described, demonstrating the functionality and applicability of the approach. Conclusions and future work are discussed.

We consider 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 edge-length. For fixed k⩾1, weprove that the first edge in the process that creates a k-connected graph coincides a.a.s. with the first edge that causes the graph to contain k/2 pairwise edge-disjoint Hamilton cycles (for even k), or (k−1)/2 Hamilton cycles plus one perfect matching, all of them pairwise edge-disjoint (for odd k). This proves and extends a conjecture of Krivelevich and M **image**ler. In the special case when k = 2, our result says that the first edge that makes the random geometric graph Hamiltonian is a.a.s. exactly the same one that gives 2-connectivity, which answers a question of Penrose. (This result appeared in three independent preprints, one of which was a precursor to this article.) We prove our results with lengths measured using the lp norm for any p>1, and we also extend our result to higher dimensions. © 2011 Wiley Periodicals, Inc. J Graph Theory 68:299-322, 2011 © 2011 Wiley Periodicals, Inc.

The authors study here low-energy linear excitations in XY and Heisenberg spin glasses at zero temperature, considering systems defined on sparse random regular graphs. The localization properties of the soft modes are linked to the spin glass transition in a field. The authors discover a new kind of low-energy excitation, called a ``Concentrated and Delocalized Low Energy Mode'' (CDLEM), given that this excitation is restricted to a sub-extensive number of vertices spread over the largest possible distances.

The genus of the binomial random graph $G(n,p)$ is well understood for a wide range of $p=p(n)$. Recently, the study of the genus of the random bipartite graph $G(n_1,n_2,p)$, with partition classes of size $n_1$ and $n_2$, was initiated by Mohar and Ying, who showed that when $n_1$ and $n_2$ are comparable in size and $p=p(n_1,n_2)$ is significantly larger than $(n_1n_2)^{-\frac{1}{2}}$ the genus of the random bipartite graph has a similar behaviour to that of the binomial random graph. In this paper we show that there is a threshold for planarity of the random bipartite graph at $p=(n_1n_2)^{-\frac{1}{2}}$ and investigate the genus close to this threshold, extending the results of Mohar and Ying. It turns out that there is qualitatively different behaviour in the case where $n_1$ and $n_2$ are comparable, when whp the genus is linear in the number of edges, than in the case where $n_1$ is asymptotically smaller than $n_2$, when whp the genus behaves like the genus of a sparse random graph $G(n_1,q)$ for an appropriately chosen $q=q(p,n_1,n_2)$.

Two ensembles are frequently used to model random graphs subject to constraints: the microcanonical ensemble (= hard constraint) and the canonical ensemble (= soft constraint). It is said that breaking of ensemble equivalence (BEE) occurs when the specific relative entropy of the two ensembles does not vanish as the size of the graph tends to infinity. Various examples have been analysed in the literature. It was found that BEE is the rule rather than the exception for two classes of constraints: sparse random graphs when the number of constraints is of the order of the number of vertices, and dense random graphs when there are two or more constraints that are frustrated. We establish BEE for a third class: dense random graphs with a single constraint on the density of a given simple graph. We show that BEE occurs in a certain range of choices for the density and the number of edges of the simple graph, which we refer to as the BEE-phase. We also show that, in part of the BEE-phase, there is a gap between the scaling limits of the averages of the maximal eigenvalue of the adjacency matrix of the random graph under the two ensembles, a property that is referred to as the spectral signature of BEE. We further show that in the replica symmetric region of the BEE-phase, BEE is due to the coexistence of two densities in the canonical ensemble.

ABSTRACTAn independent set of size in a finite undirected graph is a set of vertices of the graph, no two of which are connected by an edge. Let be the number of independent sets of size in the graph and let . In 1987, Alavi, Malde, Schwenk, and Erdős asked if the independent set sequence of a tree is unimodal (the sequence goes up and then down). This problem is still open. In 2006, Levit and Mandrescu showed that the last third of the independent set sequence of a tree is decreasing. We show that the first 46.8% of the independent set sequence of a random tree is increasing with (exponentially) high probability as the number of vertices goes to infinity. So, the question of Alavi, Malde, Schwenk, and Erdős is “four‐fifths true,” with high probability. We also show unimodality of the independent set sequence of Erdős–Rényi random graphs, when the expected degree of a single vertex is large (with [exponentially] high probability as the number of vertices in the graph goes to infinity, except for a small region near the mode). A weaker result is shown for random regular graphs. The structure of independent sets of size as varies is of interest in probability, statistical physics, combinatorics, and computer science.

We study Maker‐Breaker games played on the edge set of a random graph. Specifically, we analyze the moment a typical random graph process first becomes a Maker's win in a game in which Maker's goal is to build a graph which admits some monotone increasing property \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \begin{align*}\mathcal{P}\end{align*} \end{document}. We focus on three natural target properties for Maker's graph, namely being k ‐vertex‐connected, admitting a perfect matching, and being Hamiltonian. We prove the following optimal hitting time results: with high probability Maker wins the k ‐vertex connectivity game exactly at the time the random graph process first reaches minimum degree 2k; with high probability Maker wins the perfect matching game exactly at the time the random graph process first reaches minimum degree 2; with high probability Maker wins the Hamiltonicity game exactly at the time the random graph process first reaches minimum degree 4. The latter two statements settle conjectures of Stojaković and Szabó. We also prove generalizations of the latter two results; these generalizations partially strengthen some known results in the theory of random graphs. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011

ABSTRACTWe study the Maker‐Breaker k‐clique game played on the edge set of the random graph G(n, p). In this game, two players, Maker and Breaker, alternately claim unclaimed edges of G(n, p), until all the edges are claimed. Maker wins if he claims all the edges of a k‐clique; Breaker wins otherwise. We determine that the threshold for the graph property that Maker can win this game is at , for all k > 3, thus proving a conjecture from Ref. [Stojaković and Szabó, Random Struct Algor 26 (2005), 204–223]. More precisely, we conclude that there exist constants such that when the game is Maker's win a.a.s., and when it is Breaker's win a.a.s.For the triangle game, when k = 3, we give a more precise result, describing the hitting time of Maker's win in the random graph process. We show that, with high probability, Maker can win the triangle game exactly at the time when a copy of K5 with one edge removed appears in the random graph process. As a consequence, we are able to give an expression for the limiting probability of Maker's win in the triangle game played on the edge set of G(n, p). © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 45, 318–341, 2014