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Limiting distributions are derived for the sparse connected components that are present when a random graph on n vertices has approximately f n edges. In particular, we show that such a graph consists entirely of trees, unicyclic components, and bicyclic components with probability approaching cosh i= 0.9325 as n + m. The limiting probability that it consists o f trees, unicyclic components, and at most one another component is approximately 0.9957; the limiting probability that it is planar lies between 0.987 and 0.9998. When a random graph evolves and the number of edges passes 4n, its components grow in cyclic complexity according to an interesting Markov process whose asymptotic structure is derived. The probability that there never is more than a single component with more edges than vertices, throughout the evolution, approaches 5 wl18 = 0.8727. A "uniform" model of random graphs, which allows self-loops and multiple edges, is shown to lead to formulas that are substantially simpler than the analogous formulas for the classical random graphs of ErdBs and RCnyi. The notions of "excess" and "deficiency," which are significant characteristics of the generating function as well as of the graphs themselves, lead to a 233 the multigraph process, because it can generate graphs with self-loops x-x, and it can also generate multiple edges. Notice that a self-loop x-x is generated with probability 1 ln', while an edge x-y with x # y is generated with probability 2 ln' because it can occur either as ( x , y ) or ( y, x ) .The second evolution procedure, introduced by Erdos and Rknyi [12], is called the permutation model or the graph process. In this case we consider all N = ( ) possible edges x-y with x < y and introduce them in random order, with all N! permutations considered equally likely. In this model there are no self-loops or multiple edges.A multigraph M on n labeled vertices can be defined by a symmetric n X n matrix of nonnegative integers m x y , where mxy = myx is the number of undirected edges x-y in G. For purposes of analysis, we shall assign a compensation factor to M ; if m = Ez=, E:=, mxy is the total number of edges, the number of sequences ( x , , y , ) ( x 2 , y z ) . . . ( x , , y, ) that lead to M is then exactly (The factor 2" accounts for choosing either ( x , y ) or ( y , x ) ; the 2mxx in the denominator of K ( M ) compensates for the case x = y . The other factor m ! accounts for permutations of the pairs, with mxy! in K ( M ) to compensate for permutations between multiple edges.) Equation (1.2) tells us that K ( M ) is a natural weighting factor for a multigraph M , because it corresponds to the relative frequency with which M tends to occur in applications. For example, consider multigraphs on three vertices (1, 2, 3) having exactly three edges. The edges will form the cycle M , = {1-2, 2-3, 3-1) much more often than they will form three identical self-loops M2 = { 1-1, 1-1, 1-1}, when the multigraphs are generated in a uniform way. For if we consider the 36 possible sequences ( x , , y...

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ABSTRACT:We study the average performance of a simple greedy algorithm for finding a matching in a sparse random graph G , where c ) 0 is constant. The algorithm was first n, c r n w proposed by Karp and Sipser Proceedings of the Twenty-Second Annual IEEE Symposium on x Foundations of Computing, 1981, pp. 364᎐375 . We give significantly improved estimates of the errors made by the algorithm. For the subcritical case where c -e we show that the algorithm finds a maximum matching with high probability. If c ) e then with high probability the algorithm produces a matching which is within n 1r5qoŽ1. of maximum size.

ABSTRACT:We consider the problem of partitioning n randomly chosen integers between 1 and 2 m into two subsets such that the discrepancy, the absolute value of the difference of their sums, is minimized. A partition is called perfect if the optimum discrepancy is 0 when the sum of all n integers in the original set is even, or 1 when the sum is odd. Parameterizing the random problem in terms of κ = m/n, we prove that the problem has a phase transition at κ = 1, in the sense that for κ < 1, there are many perfect partitions with probability tending to 1 as n → ∞, whereas for κ > 1, there are no perfect partitions with probability tending to 1. Moreover, we show that this transition is first-order in the sense the derivative of the so-called entropy is discontinuous at κ = 1.We also determine the finite-size scaling window about the transition point: κ n = 1 − 2n −1 log 2 n + λ n /n, by showing that the probability of a perfect partition tends to 1 0, or some explicitly computable p λ ∈ 0 1 , depending on whether λ n tends to −∞ ∞, or λ ∈ −∞ ∞ , respectively. For λ n → −∞ fast enough, we show that the number of perfect partitions is Gaussian in the limit. For λ n → ∞, we prove that with high probability the optimum partition is unique, and that the optimum discrepancy is 2 λ n . Within the window, i.e., if λ n is bounded, we prove that the optimum discrepancy is bounded. Both for λ n → ∞ and within the window, we find the limiting distribution of the (scaled) discrepancy. Finally, both for the integer partitioning problem and for the continuous partitioning problem, we find the joint distribution of the k smallest discrepancies above the scaling window.

A process of growing a random recursive tree T,, is studied. The sequence { T,,} is shown to be a sequence of "snapshots" of a Crump-Mode branching process. This connection and a theorem by Kingman are used to show quickly that the height of T,, is asymptotic, with probability one, to c log n. In particular, c = e = 2.718. . . for the uniform recursive tree, and c = ( 2 y ) -' , where ye'+Y = 1, for the ordered recursive tree. An analogous reduction provides a short proof of Devroye's limit law for the height of a random rn-ary search tree. We show finally a close connection between another Devroye's result, on the height of a random union-find tree, and our theorem on the height of the uniform recursive tree. 0

ABSTRACT:The k-parameter bootstrap percolation on a graph is a model of an interacting particle system, which can also be viewed as a variant of a cellular automaton growth process with threshold k ≥ 2. At the start, each of the graph vertices is active with probability p and inactive with probability 1 − p, independently of other vertices. Presence of active vertices triggers a bootstrap percolation process controlled by a recursive rule: an active vertex remains active forever, and a currently inactive vertex becomes active when at least k of its neighbors are active. The basic problem is to identify, for a given graph, p − and p + such that for p < p − ( p > p + resp.) the probability that all vertices are eventually active is very close to 0 (1 resp.). The bootstrap percolation process is a deterministic process on the space of subsets of the vertex set, which is easy to describe but hard to analyze rigorously in general. We study the percolation on the random d-regular graph, d ≥ 3, via analysis of the process on the multigraph counterpart of the graph. Here, thanks to a "principle of deferred decisions," the percolation dynamics is described by a surprisingly simple Markov chain. Its generic state is formed by the counts of currently active and nonactive vertices having various degrees of activation capabilities. We replace the chain by a deterministic dynamical system, and use its integrals to show-via exponential supermartingales-that the percolation process undergoes relatively small fluctuations around the deterministic trajectory. This allows us to show existence of the phase transitionis of order n −1/2 for k < d − 1, and n −εn , (ε n ↓ 0, ε n log n → ∞), for k = d − 1.Note that p * is the same as the critical probability of the process on the corresponding infinite regular tree.

The theme of this work is an "inside-out" approach to the enumeration of graphs. It is based on a well-known decomposition of a graph into its 2-core, i.e. the largest subgraph of minimum degree 2 or more, and a forest of trees attached. Using our earlier (asymptotic) formulae for the total number of 2-cores with a given number of vertices and edges, we solve the corresponding enumeration problem for the connected 2-cores. For a subrange of the parameters, we also enumerate those 2-cores by using a deeper inside-out notion of a kernel of a connected 2-core.Using this enumeration result in combination with Caley's formula for forests, we obtain an alternative and simpler proof of the asymptotic formula of Bender, Canfield and McKay for the number of connected graphs with n vertices and m edges, with improved error estimate for a range of m values.As another application, we study the limit joint distribution of three parameters of the giant component of a random graph with n vertices in the supercritical phase, when the difference between average vertex degree and 1 far exceeds n −1/3 . The three parameters are defined in terms of the 2-core of the giant component, i.e. its largest subgraph of minimum degree 2 or more. They are the number of vertices in the 2-core, the excess (#edges − #vertices) of the 2-core, and the number of vertices not in the 2-core. We show that the limit distribution is jointly Gaussian throughout the whole supercritical phase. In particular, for the first time, the 2-core size is shown to be asymptotically normal, in the widest possible range of the average vertex degree.