Résumé : We show that by restricting the degrees of the vertices of a graph to an arbitrary set Δ, the threshold α(Δ) of the phase transition for a random graph with n vertices and m = α(Δ).n edges can be either accelerated (e.g., α(Δ) approx 0.38 for Δ = {0,1,4,5}) or postponed (e.g., α(Δ) approx 0.95 for Δ={1,2,50}) compared to a classical Erdős–Rényi random graph where α(N)=1/2. We investigate different graph statistics inside the critical window of transition (planarity, diameter, longest path...). We apply our results to a 2-SAT model with restricted literal degrees: the number of clauses that each literal is incident to belongs to the set Δ. We prove a lower bound for the probability that a formula with n variables and m=2.α(Δ) n clauses is satisfiable. This probability is close to 1 for the subcritical regime m=2.α.n.(1-μ.n^{-1/3}), μ to ∞ and improves/generalizes the lower bound of Bollobás, Borgs, Chayes, Kim, and Wilson. This shows how the phase transition threshold for 2-SAT moves if we change the degrees of the literals. Joint work with Vlady Ravelomanana.
[arXiv]
Dernière modification : Thursday 23 March 2017 | Contact : Cyril.Banderier at lipn.univ-paris13.fr |