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 (and periodically updated) directions, and an edge exists between two vertices if their Euclidean distance is below a given threshold. We provide precise asymptotic results for the expected length of the connectivity and disconnectivity periods of the network. We believe the formal tools developed in this work could be of use in future studies in more concrete settings, in the same manner as the development of connectivity threshold for static random geometric graphs has affected a lot of research done on ad hoc networks. In the process of proving results for the dynamic case we also obtain asymptotically precise bounds for the probability of the existence of a component of fixed size ℓ, ℓ ≥ 2, for the static case. La distinció per a la promoció de la recerca de la Generalitat de Catalunya, 2002. 1 connected (see Section 2). We denote this value of r by r c . Thereafter, hundreds of researchers have used those basic results on connectivity to design algorithms for more efficient coverage, communication and energy savings in ad hoc networks, and in particular for sensor networks (see the previously mentioned books). On the other hand, much work has been done on the graph theoretic properties of static RGG, comprehensively summarized in the monograph of M. D. Penrose [15]. In Section 2, we prove a result on static random geometric graphs, which was not known before (Theorem 1): At the threshold of connectivity r c and for any fixed ℓ > 1, the probability of having some component of size at least ℓ other than the giant component is asymptotically Θ(1/ log ℓ−1 n). Moreover, the most common of such components are cliques with exact size ℓ. This result plays an important role in the derivation of the main result for the dynamic setting, which is explained below.Recently, there has been an increasing interest for MANETs (mobile ad hoc networks). Several models of mobility have been proposed in the literature -for an excellent survey of those models we refer to [10]. In all these models, the connections in the network are created and destroyed as the vertices move closer together or further apart. In all previous work, the authors performed empirical studies on network topology and routing performance. The paper [5] also deals with the problem of maintaining connectivity of mobile vertices communicating by radio, but from an orthogonal perspective to the one in the present paper -it describes a kinetic data structure to maintain the connected components of the union of unit-radius disks moving in the plane.The particular mobility model we are using here (in the literature it is often called the Random Walk model) was introduced by Guerin [6], and it can be seen as the foundation for most of the mobility models developed afterwards [10]. In the Random Walk model, each vertex selects uniformly at random a direction (angle) in which to travel. The vertices s...

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) log(k − 1), then the value k − 1 is discarded and thus the chromatic number is exactly determined. Hence we improve a recently announced result by Achlioptas and Moore in which the chromatic number was allowed to take the value k + 1. Our proof applies the small subgraph conditioning method to the number of balanced k-colourings, where a colouring is balanced if the number of vertices of each colour is equal.celebrated result by Bollobás [7] later extended by Luczak [14] showed that if pn → ∞ then asymptotically almost surely (a.a.s.) χ(G(n, p)) ∼ n log(1/(1 − p)) 2 log(np) .Here and in similar statements, an event occurs a.a.s. if its probability tends to 1 as n tends to infinity. For p = c/n, Achlioptas and Naor [3] proved that the chromatic number of G(n, p) is a.a.s. k or k + 1 where k is the smallest positive integer with 2k log k > c. Moreover, they discarded the case k for roughly half of the values of c. In the same direction, Coja-Oghlan, Panagiotou and Steger [8] showed that a.a.s. χ(G(n, p)) ∈ {k, k + 1, k + 2} for p < n −3/4−ǫ where k is the smallest positive integer satisfying 2k log k > p(n − 1). Meanwhile, some other results gave concentration of the chromatic number without determining the values so precisely: Luczak [15] proved that χ(G(n, p)) is a.a.s. 2-point-concentrated if p < n −5/6−ǫ , and later Alon and Krivelevich [4] extended this to p < n −1/2−ǫ . More recently, results have been published about the chromatic number for the model G n,d of random d-regular graphs, which is the probability space on d-regular graphs with n vertices having uniform distribution. For basic results and notation on random regular graphs, see [21]. Hereinafter, dn is always assumed to be even for feasibility. For fixed d, Molloy and Reed [16] showed that if q(1 − 1/q) d/2 < 1 then χ(G n,d ) > q a.a.s. Then, for d < n 1/3−ǫ , Frieze and Luczak [11] established thatand later Cooper, Frieze, Reed and Riordan [9] extended the same asymptotic formula to apply to d ≤ n 1−ǫ . Similarly, the range n 6/7+ǫ ≤ d ≤ 0.9n was covered by Krivelevich, Sudakov, Vu and Wormald [13], who showed that χ(G n,d ) ∼ n 2 log b d a.a.s. where b = n/(n − d). Achlioptas and Moore [2] recently announced a significant new result for constant d. They stated that if k is the smallest integer satisfying d < 2(k − 1) log(k − 1) then a.a.s. χ(G n,d ) is k − 1, k, or k + 1. If, in addition, d > (2k − 3) log(k − 1), then a.a.s. χ(G n,d ) is k or k + 1. Finally, Ben-Shimon and Krivelevich [5] established 2-point concentration of χ(G n,d ) for d = o(n 1/5 ).In this paper we restrict the set of possible values for the chromatic number given by Achlioptas and Moore, and show that χ(G n,d ) a.a.s. ca...

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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 2-connectivity. We also extend this result to arbitrary connectivity, by proving 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).

Given any two vertices u, v of a random geometric graph, denote by dE(u, v) their Euclidean 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 above the connectivity threshold). Our result also improves the best known estimates on the diameter of random geometric graphs. We also provide a lower bound on dG(u, v) in terms of dE (u, v).