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It was only recently shown by Shi and Wormald, using the differential equation method to analyse an appropriate algorithm, that a random 5-regular graph asymptoti-cally almost surely has chromatic number at most 4. Here, we show that the chromatic number of a random 5-regular graph is asymptotically almost surely equal to 3, provided a certain four-variable… (More)

We present a model of the establishment and maintenance of communication between mobile agents. We assume that the agents move through a fixed environment modeled by a motion graph and are able to communicate if they are within distance at most d of each other. As the agents move randomly, we analyze the evolution in time of the connectivity between a set… (More)

We show for an arbitrary ℓp norm that the property that a random geometric graph G(n, r) contains a Hamiltonian cycle exhibits a sharp threshold at r = r(n) = log n αp n , where αp is the area of the unit disk in the ℓp norm. The proof is constructive and yields a linear time algorithm for finding a Hamiltonian cycle of G(n, r) a.a.s., provided r = r(n) ≥… (More)

We show that uniformly random 5-regular graphs of n ver-tices are 3-colorable with probability that is positive independently of n.

In this work we introduce Dynamic Random Geometric Graphs as a basic rough model for mobile wireless sensor networks, where communication distances are set to the known threshold for connectivity of static random geometric graphs. We provide precise asymptotic results for the expected length of the connectivity and disconnectivity periods of the network. We… (More)

We present a mathematical model to analyse the establishment and maintenance of communication between mobile agents. We assume that the agents move through a fixed environment modelled by a motion graph, and are able to communicate if they are at distance at most d. As the agents move randomly, we analyse the evolution in time of the connectivity between a… (More)

In this work we give precise asymptotic expressions on the probability of the existence of fixed-size components at the threshold of connectivity for random geometric graphs.

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