Xavier Pérez-Giménez

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We consider the effect on the length of the game of Cops and Robbers when more cops are added to the game play. In Overprescribed Cops and Robbers, as more cops are added, the capture time (the minimum length of the game assuming optimal play) monotonically decreases. We give the full range of capture times for any number of cops on trees, and classify the(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 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 show that a randomly chosen 3-CNF formula over n variables with clauses-tovariables 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 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 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)
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 study the arboricity A and the maximum number T of edge-disjoint spanning trees of the Erdős-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 bm/(n−1)c, where δ is the smallest degree of the graph and m denotes the number of edges. Moreover, we explicitly determine a sharp(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)