A clique coloring of a graph is a coloring of the vertices so that no maximal clique is monochromatic (ignoring isolated vertices). The smallest number of colors in such a coloring is the clique chromatic number. In this paper, we study the asymptotic behavior of the clique chromatic number of the random graph 𝒢(n,p) for a wide range of edge‐probabilities p = p(n). We see that the typical clique chromatic number, as a function of the average degree, forms an intriguing step function.

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...

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Random hyperbolic graphs were recently introduced by Krioukov et. al. [KPK + 10] as a model for large networks. Gugelmann, Panagiotou, and Peter [GPP12] then initiated the rigorous study of random hyperbolic graphs using the following model: for α > 1 2 , C ∈ R, n ∈ N, set R = 2 ln n + C and build the graph G = (V, E) with |V | = n as follows: For each v ∈ V , generate i.i.d. polar coordinates (rv, θv) using the joint density function f (r, θ), with θv chosen uniformly from [0, 2π) and rv with density f (r) = α sinh(αr) cosh(αR)−1 for 0 ≤ r < R. Then, join two vertices by an edge, if their hyperbolic distance is at most R. We prove that in the range ), thus answering a question raised in [GPP12] concerning the diameter of such random graphs. As a corollary from our proof we obtain that the second largest component has size O(log 2C 0 +1+o(1) n), thus answering a question of Bode, Fountoulakis and Müller [BFM13]. We also show that a.a.s. there exist isolated components forming a path of length Ω(log n), thus yielding a lower bound on the size of the second largest component.

The metric dimension of a graph G is the minimum number of vertices in a subset S of the vertex set of G such that all other vertices are uniquely determined by their distances to the vertices in S. In this paper we investigate the metric dimension of the random graph G(n, p) for a wide range of probabilities p = p(n).1991 Mathematics Subject Classification. 05C12, 05C35, 05C80.