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We show that a randomly chosen 3-CNF formula over n variables with clauses-to-variables ratio at least 4.4898 is asymptotically almost surely unsatisfiable. The previous best such bound, due to Dubois in 1999, was 4.506. The first such bound, independently discovered by many groups of researchers since 1983, was 5.19. Several decreasing values between 5.19… (More)

In this paper we present a new upper bound for randomly chosen 3-CNF formulas. In particular we show that any random formula over n variables, with a clauses-to-variables ratio of at least 4.4898 is, as n grows large, asymptotically almost surely unsatisfiable. The previous best such bound, due to Dubois in 1999, was 4.506. The first such bound,… (More)

We provide the first rigorous analytical results for the connectivity of dynamic random geometric graphs - a model for mobile wireless networks in which vertices move in random directions in the unit torus. The model presented here follows the one described. We provide precise asymptotic results for the expected length of the connectivity and… (More)

We provide the first analytical results for the connectivity of dynamic random geometric graphs --- a model of mobile wireless networks in which vertices move in random directions, and an edge exists between two vertices if their Euclidean distance is below a given value. We provide precise asymptotic results for the expected length of the connectivity and… (More)

In this work we show that, for any fixed d, random d-regular graphs asymptotically almost surely can be coloured with k colours, where k is the smallest integer satisfying d < 2(k − 1) log(k − 1). From previous lower bounds due to Molloy and Reed, this establishes the chromatic number to be asymptotically almost surely k − 1 or k. If moreover d > (2k − 3)… (More)

We study the arboricity A and the maximum number T of edge-disjoint spanning trees of the Erd˝ os-Rényi random graph G (n, p). For all p(n) ∈ [0, 1], we show that, with high probability, T is precisely the minimum between δ and m/(n − 1), where δ is the smallest degree of the graph and m denotes the number of edges. Moreover, we explicitly determine a sharp… (More)

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 p norm. We show that the first edge that makes the random geometric graph Hamiltonian is a.a.s. exactly the same one that gives… (More)

Given any two vertices u, v of a random geometric graph, denote by dE(u, v) their Eu-clidean distance and by dG(u, v) their graph distance. The problem of finding upper bounds on dG(u, v) in terms of dE(u, v) has received a lot of attention in the literature [1, 2, 6, 8]. In this paper, we improve these upper bounds for values of r = ω(√ log n) (i.e. for r… (More)

We consider the domination number for on-line social networks, both in a stochastic network model, and for real-world, networked data. Asymptotic sublinear bounds are rigorously derived for the domination number of graphs generated by the memoryless geometric protean random graph model. We establish sublinear bounds for the domination number of graphs in… (More)