We prove the existence of a large complete subgraph w.h.p. in a preferential attachment random graph process with an edge-step. That is, we consider a dynamic stochastic process for constructing a graph in which at each step we independently decide, with probability p ∈ (0, 1), whether the graph receives a new vertex or a new edge between existing vertices. The connections are then made according to a preferential attachment rule. We prove that the random graph G t produced by this so-called GLP (Generalized linear preferential) model at time t contains a complete subgraph whose vertex set cardinality is given by t α , where α = (1 − ε) 1−p 2−p , for any small ε > 0 asymptotically almost surely.
In the framework of the probabilistic method in combinatorics we present a new local lemma based on an entropy compression random algorithm and we show through examples that it can efficiently replace the Moser-Tardos algorithmic version of the Lovász Local Lemma as well as its Pegden extension in a restricted setting where the underlying probability space of the bad events is generated by a finite family of i.i.d. uniform random variables. We then use this new lemma to improve the upper bound for the β-frugal vertex chromatic index of a bounded degree graph recently obtained by Ndreca et al. (Eur J Comb 33: 592-609, 2012).
In the Constrained-degree percolation model on a graph (V, E) there are a sequence, (U e ) e∈E , of i.i.d. random variables with distribution U [0, 1] and a positive integer k. Each bond e tries to open at time U e , it succeeds if both its end-vertices would have degrees at most k − 1. We prove a phase transition theorem for this model on the square lattice L 2 , as well as on the d-ary regular tree. We also prove that on the square lattice the infinite cluster is unique in the supercritical phase.
We consider the Bernoulli bond percolation process (with parameter p) on infinite graphs and we give a general criterion for bounded degree graphs to exhibit a non-trivial percolation threshold based either on a single isoperimetric inequality if the graph has a bi-infinite geodesic, or two isoperimetric inequalities if the graph has not a bi-infinite geodesic. This new criterion extends previous criteria and brings together a large class of amenable graphs (such as regular lattices) and non-amenable graphs (such trees). We also study the finite connectivity in graphs satisfying the new general criterion and show that graphs in this class with a bi-infinite geodesic always have finite connectivity functions with exponential decay as p is sufficiently close to one. On the other hand, we show that there are graphs in the same class with no bi-infinite geodesic for which the finite connectivity decays sub-exponentially (down to polynomially) in the highly supercritical phase even for p arbitrarily close to one.
In this work we investigate a preferential attachment model whose parameter is a function f : N → [0, 1] that drives the asymptotic proportion between the numbers of vertices and edges of the graph. We investigate topological features of the graphs, proving general bounds for the diameter and the clique number. Our results regarding the diameter are sharp when f is a regularly varying function at infinity with strictly negative index of regular variation −γ. For this particular class, we prove a characterization for the diameter that depends only on −γ. More specifically, we prove that the diameter of such graphs is of order 1/γ with high probability, although its vertex set order goes to infinity polynomially. Sharp results for the diameter for a wide class of slowly varying functions are also obtained. The almost sure convergence for the properly normalized logarithm of the clique number of the graphs generated by slowly varying functions is also proved.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.