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- Anthony Bonato, Xavier Pérez-Giménez, Pawel Pralat, Benjamin Reiniger
- Graphs and Combinatorics
- 2017

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)

- Josep Díaz, Lefteris M. Kirousis, Dieter Mitsche, Xavier Pérez-Giménez
- FSTTCS
- 2008

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)

- Josep Díaz, Lefteris M. Kirousis, Dieter Mitsche, Xavier Pérez-Giménez
- Theor. Comput. Sci.
- 2009

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)

- Josep Díaz, Dieter Mitsche, Xavier Pérez-Giménez
- SODA
- 2008

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)

- Josep Díaz, Dieter Mitsche, Xavier Pérez-Giménez
- IEEE Transactions on Mobile Computing
- 2009

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)

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)

- Pu Gao, Xavier Pérez-Giménez, Cristiane M. Sato
- SODA
- 2014

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)

- Konstantinos Panagiotou, Xavier Pérez-Giménez, Thomas Sauerwald, He Sun
- Combinatorics, Probability & Computing
- 2015

- Tobias Müller, Xavier Pérez-Giménez, Nicholas C. Wormald
- Journal of Graph Theory
- 2011

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)

- Josep Díaz, Dieter Mitsche, Guillem Perarnau, Xavier Pérez-Giménez
- ArXiv
- 2014

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)