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We show that for many models of random trees, the independence number divided by the size converges almost surely to a constant as the size grows to infinity; the trees that we consider include random recursive trees, binary and $m$-ary search trees, preferential attachment trees, and others. The limiting constant is computed, analytically or numerically, for several examples. The method is based on Crump-Mode-Jagers branching processes., 15 pages (v2: reference added)

Divide-and-conquer recurrences of the form f ( n ) = f (⌊ n/2⌋ ) + f ( ⌈ n/2⌉ ) + g ( n ) ( n ⩾ 2), with g ( n ) and f (1) given, appear very frequently in the analysis of computer algorithms and related areas. While most previous methods and results focus on simpler crude approximation to the solution, we show that the solution always satisfies the simple identity f ( n ) = n P (log 2 n ) − Q ( n ) under an optimum (iff) condition on g ( n ). This form is not only an identity but also an asymptotic expansion because Q ( n ) is of a smaller order than linearity. Explicit forms for the continuous periodic function P are provided. We show how our results can be easily applied to many dozens of concrete examples collected from the literature and how they can be extended in various directions. Our method of proof is surprisingly simple and elementary but leads to the strongest types of results for all examples to which our theory applies.

We analyse the size of an independent set in a random graph on $n$ vertices with specified vertex degrees, constructed via a simple greedy algorithm: order the vertices arbitrarily, and, for each vertex in turn, place it in the independent set unless it is adjacent to some vertex already chosen. We find the limit of the expected proportion of vertices in the greedy independent set as $n \to \infty$, expressed as an integral whose upper limit is defined implicitly, valid whenever the second moment of a random vertex degree is uniformly bounded. We further show that the random proportion of vertices in the independent set converges to the jamming constant as $n \to \infty$. The results hold under weaker assumptions in a random multigraph with given degrees constructed via the configuration model., Comment: 25 pages

We give an example of a graphon such that there is no equivalent graphon with a degree function that is (weakly) increasing., 4 pages

We consider the generalization of the P\'olya urn scheme with possibly infinite many colors as introduced in \cite{Th-Thesis, BaTH2014, BaTh2016, BaTh2017}. For countable many colors, we prove almost sure convergence of the urn configuration under \emph{uniform ergodicity} assumption on the associated Markov chain. The proof uses a stochastic coupling of the sequence of chosen colors with a \emph{branching Markov chain} on a weighted \emph{random recursive tree} as described in \cite{BaTh2017, Sv_2018}. Using this coupling we estimate the covariance between any two selected colors. In particular, we reprove the limit theorem for the classical urn models with finitely many colors., Comment: 13 pages

We consider the general version of Pólya urns recently studied by Bandyopadhyay and Thacker (2016+) and Mailler and Marckert (2017), with the space of colours being any Borel space $S$ and the state of the urn being a finite measure on $S$. We consider urns with random replacements, and show that these can be regarded as urns with deterministic replacements using the colour space $S\times [0,1]$.

A random vector $X=(X_1,\ldots,X_n)$ with the $X_i$ taking values in an arbitrary measurable space $(S, \mathscr{S})$ is exchangeable if its law is the same as that of $(X_{\sigma(1)}, \ldots, X_{\sigma(n)})$ for any permutation $\sigma$. We give an alternative and shorter proof of the representation result (Jaynes \cite{Jay86} and Kerns and Sz\'ekely \cite{KS06}) stating that the law of $X$ is a mixture of product probability measures with respect to a signed mixing measure. The result is "finitistic" in nature meaning that it is a matter of linear algebra for finite $S$. The passing from finite $S$ to an arbitrary one may pose some measure-theoretic difficulties which are avoided by our proof. The mixing signed measure is not unique (examples are given), but we pay more attention to the one constructed in the proof ("canonical mixing measure") by pointing out some of its characteristics. The mixing measure is, in general, defined on the space of probability measures on $S$, but for $S=\mathbb{R}$, one can choose a mixing measure on $\mathbb{R}^n$., Comment: We here give an alternative proof of the measurability of the random signed-measure underlying the construction. We also add an independent proof of the main algebraic fact used in the paper. Title updated

We consider a random permutation drawn from the set of permutations of length $n$ that avoid some given set of patterns of length 3. We show that the number of occurrences of another pattern $\sigma$ has a limit distribution, after suitable scaling. In several cases, the number is asymptotically normal; this contrasts to the cases of permutations avoiding a single pattern of length 3 studied in earlier papers., Comment: 23 pages

We find Feynman–Kac type representation theorems for generalized diffusions. To do this we need to establish existence, uniqueness and regularity results for equations with measure-valued coefficients.

We study four different notions of convergence for graphexes, recently introduced by Borgs, Chayes, Cohn and Holden, and by Veitch and Roy. We give some properties of them and some relations between them. We also extend results by Veitch and Roy on convergence of empirical graphons., 16 pages. To appear

We explore the size of the largest (permuted) triangular submatrix of a random matrix, and more precisely its asymptotical behavior as the size of the ambient matrix tends to infinity. The importance of such permuted triangular submatrices arises when dealing with certain combinatorial algebraic settings in which these submatrices determine the rank of the ambient matrix, and thus attract a special attention., 11 pages

We consider conditioned Galton-Watson trees and show asymptotic normality of additive functionals that are defined by toll functions that are not too large. This includes, as a special case, asymptotic normality of the number of fringe subtrees isomorphic to any given tree, and joint asymptotic normality for several such subtree counts. Another example is the number of protected nodes. The offspring distribution defining the random tree is assumed to have expectation 1 and finite variance; no further moment condition is assumed., 49 pages

The theory of influence and sharp threshold is a key tool in probability and probabilistic combinatorics, with numerous applications. One significant aspect of the theory is directed at identifying the level of generality of the product probability space that accommodates the event under study. We derive the influence inequality for a completely general product space, by establishing a relationship to the Lebesgue cube studied by Bourgain, Kahn, Kalai, Katznelson and Linial (BKKKL) in 1992. This resolves one of the assertions of BKKKL. Our conclusion is valid also in the setting of the generalized influences of Keller.

This survey studies asymptotics of random fringe trees and extended fringe trees in random trees that can be constructed as family trees of a Crump-Mode-Jagers branching process, stopped at a suitable time. This includes random recursive trees, preferential attachment trees, fragmentation trees, binary search trees and (more generally) $m$-ary search trees, as well as some other classes of random trees. We begin with general results, mainly due to Aldous (1991) and Jagers and Nerman (1984). The general results are applied to fringe trees and extended fringe trees for several particular types of random trees, where the theory is developed in detail. In particular, we consider fringe trees of $m$-ary search trees in detail; this seems to be new. Various applications are given, including degree distribution, protected nodes and maximal clades for various types of random trees. Again, we emphasise results for $m$-ary search trees, and give for example new results on protected nodes in $m$-ary search trees. A separate section surveys results on height, saturation level, typical depth and total path length, due to Devroye (1986), Biggins (1995, 1997) and others. This survey contains well-known basic results together with some additional general results as well as many new examples and applications for various classes of random trees.

The time it takes a random walker in a lattice to reach the origin from another vertex $x$, has infinite mean. If the walker can restart the walk at $x$ at will, then the minimum expected hitting time $T(x,0)$ (minimized over restarting strategies) is finite; it was called the ``grade'' of $x$ by Dumitriu, Tetali and Winkler. They showed that, in a more general setting, the grade (a variant of the ``Gittins index'') plays a crucial role in control problems involving several Markov chains. Here we establish several conjectures of Dumitriu et al on the asymptotics of the grade in Euclidean lattices. In particular, we show that in the planar square lattice, $T(x,0)$ is asymptotic to $2|x|^2\log|x|$ as $|x| \to \infty$. The proof hinges on the local variance of the potential kernel $h$ being almost constant on the level sets of $h$. We also show how the same method yields precise second order asymptotics for hitting times of a random walk (without restarts) in a lattice disk., 12 pages

We study random graphs, both G( n,p) and G( n,m), with random orientations on the edges. For three fixed distinct vertices s,a,b we study the correlation, in the combine probability space, of the events $\{a\to s\}$ and $\{s\to b\}$ . For G(n,p), we prove that there is a $pc = 1/2$ such that for a fixed $p the correlation is negative for large enough n and for $p > pc$ the correlation is positive for large enough n. We conjecture that for a fixed $n \ge 27$ the correlation changes sign three times for three critical values of p. For G(n,m) it is similarly proved that, with $p=m/({{n}\atop {2}})$ , there is a critical pc that is the solution to a certain equation and approximately equal to 0.7993. A lemma, which computes the probability of non existence of any $\ell$ directed edges in G(n,m), is thought to be of independent interest. We present exact recursions to compute **math-image** and **math-image**. We also briefly discuss the corresponding question in the quenched version of the problem. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011 (Supported by Knut and Alice Wallenberg Foundation (Royal Swedish Academy of Sciences Research Fellow)(Svante Linusson). Part of this research was done at Institut Mittag-Leffler, Djursholm, Sweden, which we thank for their hospitality.)

Bona [2007+] studied the distribution of ascents, plateaux and descents in the class of Stirling permutations, introduced by Gessel and Stanley [1978]. Recently, Janson [2008+] showed the connection between Stirling permutations and plane recursive trees and proved a joint normal law for the parameters considered by Bona. Here we will consider generalized Stirling permutations extending the earlier results of Bona and Janson, and relate them with certain families of generalized plane recursive trees, and also $(k+1)$-ary increasing trees. We also give two different bijections between certain families of increasing trees, which both give as a special case a bijection between ternary increasing trees and plane recursive trees. In order to describe the (asymptotic) behaviour of the parameters of interests, we study three (generalized) Polya urn models using various methods., 25 pages, 3 figures, Keywords: Increasing trees, plane recursive trees, Stirling permutations, ascents, descents, urn models, limiting distributions

We consider identifying codes in binary Hamming spaces F^n, i.e., in binary hypercubes. The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in 1998. Currently, the subject forms a topic of its own with several possible applications, for example, to sensor networks. Let C be a subset of F^n. For any subset X of F^n, denote by I_r(X)=I_r(C;X) the set of elements of C within distance r from at least one x in X. Now C is called an (r, 13 pages

We study the space requirements of a sorting algorithm where only items that at the end will be adjacent are kept together. This is equivalent to the following combinatorial problem: Consider a string of fixed length n that starts as a string of 0's, and then evolves by changing each 0 to 1, with then changes done in random order. What is the maximal number of runs of 1's? We give asymptotic results for the distribution and mean. It turns out that, as in many problems involving a maximum, the maximum is asymptotically normal, with fluctuations of order n^{1/2}, and to the first order well approximated by the number of runs at the instance when the expectation is maximized, in this case when half the elements have changed to 1; there is also a second order term of order n^{1/3}. We also treat some variations, including priority queues. The proofs use methods originally developed for random graphs., Comment: 31 PAGES

We prove general limit theorems for sums of functions of subtrees of (random) binary search trees and random recursive trees. The proofs use a new version of a representation by Devroye, and Stein's method for both normal and Poisson approximation together with certain couplings. ¶ As a consequence, we give simple new proofs of the fact that the number of fringe trees of size $ k=k_n $ in the binary search tree or in the random recursive tree (of total size $ n $) has an asymptotical Poisson distribution if $ k\rightarrow\infty $, and that the distribution is asymptotically normal for $ k=o(\sqrt{n}) $. Furthermore, we prove similar results for the number of subtrees of size $ k $ with some required property $ P $, e.g., the number of copies of a certain fixed subtree $ T $. Using the Cramér-Wold device, we show also that these random numbers for different fixed subtrees converge jointly to a multivariate normal distribution. We complete the paper by giving examples of applications of the general results, e.g., we obtain a normal limit law for the number of $ \ell $-protected nodes in a binary search tree or in a random recursive tree.

We study the distribution of the individual displacements in hashing with linear probing for three different versions: First Come, Last Come and Robin Hood. Asymptotic distributions and their moments are found when the the size of the hash table tends to infinity with the proportion of occupied cells converging to some α, 0 < α < 1. (In the case of Last Come, the results are more complicated and less complete than in the other cases.)We also show, using the diagonal Poisson transform studied by Poblete, Viola and Munro, that exact expressions for finite m and n can be obtained from the limits as m,n → ∞.We end with some results, conjectures and questions about the shape of the limit distributions. These have some relevance for computer applications.

\noindent We describe the asymptotic behavior of the cardinalities of the finite symmetric inverse semigroup ISn and its endomorphism semigroup. This is applied to show that the ratio |ISn|/|End(ISn)| is asymptotically 0, answering a question of Schein and Teclezghi. We also apply our results to compute the distributions of elements from ISn with respect to certain combinatorial properties, and to compute the generating functions for |ISn| and for the number of nilpotent elements in ISn.

Random-cluster measures on infinite regular trees are studied in conjunction with a general type of `boundary condition', namely an equivalence relation on the set of infinite paths of the tree. The uniqueness and non-uniqueness of random-cluster measures are explored for certain classes of equivalence relations. In proving uniqueness, the following problem concerning branching processes is encountered and answered. Consider bond percolation on the family-tree $T$ of a branching process. What is the probability that every infinite path of $T$, beginning at its root, contains some vertex which is itself the root of an infinite open sub-tree?

LetG be a fixed graph and letX G be the number of copies ofG contained in the random graphG(n, p). We prove exponential bounds on the upper tail ofX G which are best possible up to a logarithmic factor in the exponent. Our argument relies on an extension of Alon’s result about the maximum number of copies ofG in a graph with a given number of edges. Similar bounds are proved for the random graphG(n, M) too.

We present a new method to show concentration of the upper tail of random variables that can be written as sums of variables with plenty of independence. We compare our method with the martingale method by Kim and Vu, which often leads to similar results.Some applications are given to the number X G of copies of a graph G in the random graph $$\Bbb G $$ (n,p). In particular, for G = K 4 and G = C 4 we improve the earlier known upper bounds on —ln ℙ(X G ≥ 2 $$\Bbb E$$ X G ) in some range of p = p(n).

We consider a random generalized railway defined as a random 3-regular multigraph where some vertices are regarded as switches that only allow traffic between certain pairs of attached edges. It is shown that the probability that the generalized railway is functioning is linear in the proportion of switches. Thus there is no threshold phenomenon for this property.

A "dyadic rectangle" is a set of the form R = [a2-s,(a + 1)2-s] × [b2-t,(b + 1)2-t], where s and t are nonnegative integers. A dyadic tiling is a tiling of the unit square with dyadic rectangles. In this paper we study n-tilings, which consist of 2n nonoverlapping dyadic rectangles, each of area 2-n, whose union is the unit square. We discuss some of the underlying combinatorial structures, provide some efficient methods for uniformly sampling from the set of n-tilings, and study some limiting properties of random tilings.

The space of permutation pseudographs is a probabilistic model of 2-regular pseudographs on n vertices, where a pseudograph is produced by choosing a permutation σ of {1,2,…, n} uniformly at random and taking the n edges {i,σ(i)}. We prove several contiguity results involving permutation pseudographs (contiguity is a kind of asymptotic equivalence of sequences of probability spaces). Namely, we show that a random 4-regular pseudograph is contiguous with the sum of two permutation pseudographs, the sum of a permutation pseudograph and a random Hamilton cycle, and the sum of a permutation pseudograph and a random 2-regular pseudograph. (The sum of two random pseudograph spaces is defined by choosing a pseudograph from each space independently and taking the union of the edges of the two pseudographs.) All these results are proved simultaneously by working in a general setting, where each cycle of the permutation is given a nonnegative constant multiplicative weight. A further contiguity result is proved involving the union of a weighted permutation pseudograph and a random regular graph of arbitrary degree. All corresponding results for simple graphs are obtained as corollaries.

Motivated by the problem of graph structure compression under realistic source models, we study the symmetry behavior of preferential and uniform attachment graphs. These are two dynamic models of network growth in which new nodes attach to a constant number $m$ of existing ones according to some attachment scheme. We prove symmetry results for $m=1$ and $2$, and we conjecture that for $m\geq 3$, both models yield asymmetry with high probability. We provide new empirical evidence in terms of graph defect. We also prove that vertex defects in the uniform attachment model grow at most logarithmically with graph size, then use this to prove a weak asymmetry result for all values of $m$ in the uniform attachment model. Finally, we introduce a natural variation of the two models that incorporates preference of new nodes for nodes of a similar age, and we show that the change introduces symmetry for all values of $m$.

Let $G=(V,E)$ be an undirected graph without loops and multiple edges. A subset $C\subseteq V$ is called \emph{identifying} if for every vertex $x\in V$ the intersection of $C$ and the closed neighbourhood of $x$ is nonempty, and these intersections are different for different vertices $x$. Let $k$ be a positive integer. We will consider graphs where \emph{every} $k$-subset is identifying. We prove that for every $k>1$ the maximal order of such a graph is at most $2k-2.$ Constructions attaining the maximal order are given for infinitely many values of $k.$ The corresponding problem of $k$-subsets identifying any at most $\ell$ vertices is considered as well., 21 pages

Bootstrap percolation is a type of cellular automaton which has been used to model various physical phenomena, such as ferromagnetism. For each natural number $r$, the $r$-neighbour bootstrap process is an update rule for vertices of a graph in one of two states: `infected' or `healthy'. In consecutive rounds, each healthy vertex with at least $r$ infected neighbours becomes itself infected. Percolation is said to occur if every vertex is eventually infected. Usually, the starting set of infected vertices is chosen at random, with all vertices initially infected independently with probability $p$. In that case, given a graph $G$ and infection threshold $r$, a quantity of interest is the critical probability, $p_c(G,r)$, at which percolation becomes likely to occur. In this paper, we look at infinite trees and, answering a problem posed by Balogh, Peres and Pete, we show that for any $b \geq r$ and for any $��> 0$ there exists a tree $T$ with branching number $\br(T) = b$ and critical probability $p_c(T,r) < ��$. However, this is false if we limit ourselves to the well-studied family of Galton--Watson trees. We show that for every $r \geq 2$ there exists a constant $c_r>0$ such that if $T$ is a Galton--Watson tree with branching number $\br(T) = b \geq r$ then p_c(T,r) > \frac{c_r}{b} e^{-\frac{b}{r-1}}. We also show that this bound is sharp up to a factor of $O(b)$ by giving an explicit family of Galton--Watson trees with critical probability bounded from above by $C_r e^{-\frac{b}{r-1}}$ for some constant $C_r>0$., 25 pages

We study protected nodes in $m$ - ary search trees, by putting them in context of generalised Po lya urns. We show that the number of two-protected nodes (the nodes that are neither leaves nor parents of leaves) in a random ternary search tree is asymptotically normal. The methods apply in principle to $m $ - ary search trees with larger $m$ as well, although the size of the matrices used in the calculations grow rapidly with $ m $ ; we conjecture that the method yields an asymptotically normal distribution for all $m\leq 26$ . The one-protected nodes, and their complement, i.e., the leaves, are easier to analyze. By using a simpler Po lya urn (that is similar to the one that has earlier been used to study the total number of nodes in $ m $ - ary search trees), we prove normal limit laws for the number of one-protected nodes and the number of leaves for all $ m\leq 26 $ .

We consider a relatively new combinatorial structure called staircase tableaux. They were introduced in the context of the asymmetric exclusion process and Askey–Wilson polynomials; however, their purely combinatorial properties have gained considerable interest in the past few years.

We study moments and asymptotic distributions of the construction cost, measured as the total displacement, for hash tables using linear probing. Four different methods are employed for different ranges of the parameters; together they yield a complete description. This extends earlier results by Flajolet, Poblete and Viola [On the analysis of linear probing hashing, Algorithmica 22 (1998), 490–515]. The average cost of unsuccessful searches is considered too. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19: 438–471, 2001

The creation and growth of components of a given complexity in a random graph process are studied. In particular, the expected number and total size of all such components is found. It follows that the largest '-component during the process is Op(n 2/3 ) for any given '. The results also yield a new proof of the asymptotic behaviour of Wright's coecients. between the two processes is that in {G(n,m)}, the edges are added at the fixed times 1,2,..., so at time m we have the random graph G(n,m) with m edges, while in {G(n,t)} the edges are added at random times in such a way that at time t = p, we have the random graph G(n,p). ({G(n,t)} may be constructed by letting each edge e in the complete graph Kn appear at a random time Te, with Te independent and uniformly distributed on (0,1), and letting G(n,t) contain the edges that have appeared before t.) In this paper, we will study random variables that depend on the order the edges appear in the process, but not on the time scale. All such variables will thus have the same distribution for both processes (not only asymptotically, but also for each finite n). Hence all results below are valid for both processes. We define the complexity of a connected graph to be its number of edges minus its number of vertices. A component of a graph of complexity ' is called an '-component. Here ' 1; a ( 1)-component is a tree, a 0-component is unicyclic, and '-components with ' 1 are known as complex components. In the beginning of the random graph process, there are no edges at all, and thus n components of order 1 and complexity 1; at the end, we have the complete graph, with a single component of complexity n 2 n. Several authors have studied what happens in between, see e.g. (4, 9, 6, 7, 11, 8). We will here add some results obtained by studying, as in (6), the ways '-components are created. Each time a new edge is added, there are two possibilities

Consider the minimal weights of paths between two points in a complete graph Kn with random weights on the edges, the weights being, for instance, uniformly distributed. It is shown that, asymptotically, this is log n/n for two given points, that the maximum if one point is fixed and the other varies is 2 log n/n, and that the maximum over all pairs of points is 3 log n/n.Some further related results are given as well, including results on asymptotic distributions and moments, and on the number of edges in the minimal weight paths.

We show that the variance of the number of edges in the random sphere of influence graph built on n i.i sites which are uniformly distributed over the unit cube in R-d, grows linearly with n. This is then used to establish a central limit theorem for the

A perturbation technique can be used to simplify and sharpen A. C. Yao's theorems about the behavior of shellsort with increments $(h,g,1)$. In particular, when $h=\Theta(n^{7/15})$ and $g=\Theta(h^{1/5})$, the average running time is $O(n^{23/15})$. The proof involves interesting properties of the inversions in random permutations that have been $h$-sorted and $g$-sorted.

We study the relation between the growth rate of a graph property and the entropy of the graph limits that arise from graphs with that property. In particular, for hereditary classes we obtain a new description of the colouring number, which by well-known results describes the rate of growth. We study also random graphs and their entropies. We show, for example, that if a hereditary property has a unique limiting graphon with maximal entropy, then a random graph with this property, selected uniformly at random from all such graphs with a given order, converges to this maximizing graphon as the order tends to infinity., 24 pages

We study protected nodes in various classes of random rooted trees by putting them in the general context of fringe subtrees introduced by Aldous (1991). Several types of random trees are considered: simply generated trees (or conditioned Galton-Watson trees), which includes several cases treated separately by other authors, binary search trees and random recursive trees. This gives unified and simple proofs of several earlier results, as well as new results., 11 pages

We study the random graph $G(n,p)$ with a random orientation. For three fixed vertices $s,a,b$ in $G(n,p)$ we study the correlation of the events $a \to s$ and $s\to b$. We prove that asymptotically the correlation is negative for small $p$, $p, Comment: 15 pages, 3 figures

We study statistical properties of the random variables $X_{\sigma}(\pi)$, the number of occurrences of the pattern $\sigma$ in the permutation $\pi$. We present two contrasting approaches to this problem: traditional probability theory and the ``less traditional'' computational approach. Through the perspective of the first one, we prove that for any pair of patterns $\sigma$ and $\tau$, the random variables $X_{\sigma}$ and $X_{\tau}$ are jointly asymptotically normal (when the permutation is chosen from $S_{n}$). From the other perspective, we develop algorithms that can show asymptotic normality and joint asymptotic normality (up to a point) and derive explicit formulas for quite a few moments and mixed moments empirically, yet rigorously. The computational approach can also be extended to the case where permutations are drawn from a set of pattern avoiders to produce many empirical moments and mixed moments. This data suggests that some random variables are not asymptotically normal in this setting., Comment: 18 pages

We study the susceptible-infective-recovered (SIR) epidemic on a random graph chosen uniformly subject to having given vertex degrees. In this model infective vertices infect each of their susceptible neighbours, and recover, at a constant rate. Suppose that initially there are only a few infective vertices. We prove there is a threshold for a parameter involving the rates and vertex degrees below which only a small number of infections occur. Above the threshold a large outbreak occurs with probability bounded away from zero. Our main result is that, conditional on a large outbreak, the evolutions of certain quantities of interest, such as the fraction of infective vertices, converge to deterministic functions of time. We also consider more general initial conditions for the epidemic, and derive criteria for a simple vaccination strategy to be successful. In contrast to earlier results for this model, our approach only requires basic regularity conditions and a uniformly bounded second moment of the degree of a random vertex. En route, we prove analogous results for the epidemic on the configuration model multigraph under much weaker conditions. Essentially, our main result requires only that the initial values for our processes converge, i.e. it is the best possible., Comment: 37 pages, no figures (revisewd following referee comments)

This is an erratum for EJP volume 16 paper 12 . We fix a gap in the proof of our estimates for odd moments.

The minimal weight of a spanning tree in a complete graph Kn with independent, uniformly distributed random weights on the edges is shown to have an asymptotic normal distribution. The proof uses a functional limit extension of results by Barbour and Pittel on the distribution of the number of tree components of given sizes in a random graph.

The asymptotic distribution of the number of Hamilton cycles in a random regular graph is determined. The limit distribution is of an unusual type; it is the distribution of a variable whose logarithm can be written as an infinite linear combination of independent Poisson variables, and thus the logarithm has an infinitely divisible distribution with a certain discrete Lévy measure. Similar results are found for some related problems. These limit results imply that some different models of random regular graphs are contiguous, which means that they are qualitatively asymptotically equivalent. For example, if r > 3, then the usual (uniformly distributed) random r-regular graph is contiguous to the one constructed by taking the union of r perfect matchings on the same vertex set (assumed to be of even cardinality), conditioned on there being no multiple edges. Some consequences of contiguity for asymptotic distributions are discussed.

Let E={X(1),X(2),...,X(m)} where the X(i) subset of or equal to V for 1 less than or equal to i less than or equal to m are distinct. The hypergraph G=(V,E) is said to be s-uniform if \X(1)\=s for 1 less than or equal to i less than or equal to m. A set o

A Fourier transform for (real valued) functions of graphs is denned. This is used to study and characterize some classes of graphs that arise as exceptional cases in limit theorems for subgraph counts in random graphs. © 1995 Wiley Periodicals, Inc.