Abstract: The numbers of spanning trees, Hamilton cycles and perfect matchings in a random graph Gnm are shown to be asymptotically normal if m is neither too large nor too small. At the lowest limit m ≍ n3/2, these numbers are asymptotically log-normal. For Gnp, the numbers are asymptotically log-normal for a wide range of p, including p constant. The same results are obtained for random directed graphs and bipartite graphs. The results are proved using decomposition and projection methods.

Abstract: A lattice basis bi,…,bm is called block reduced with block size β if for every β consecutive vectors bi,…,bi+β−1, the orthogonal projections of bi,…,bi+β−1 in span(bi,…,bi−1)⊥ are reduced in the sense of Hermite and Korkin–Zolotarev. Let λi denote the successive minima of lattice L, and let b1,…,bm be a basis of L that is block reduced with block size β. We prove that for i = 1,…, mwhere γβ is the Hermite constant for dimension β. For block size β = 3 and odd rank m ≥ 3, we show thatwhere the maximum is taken over all block reduced bases of all lattices L. We present critical block reduced bases achieving this maximum. Using block reduced bases, we improve Babai's construction of a nearby lattice point. Given a block reduced basis with block size β of the lattice L, and given a point x in the span of L, a lattice point υ can be found in time βO(β) satisfyingThese results also give improvements on the method of solving integer programming problems via basis reduction.

Abstract: A simple proof is given of the best-known upper bound on the cardinality of a set of vectors of length t over an alphabet of size b, with the property that, for every subset of k vectors, there is a coordinate in which they all differ This question is motivated by the study of perfect hash functions

Abstract: The main result of this paper is that for every 2 ≤ r s , and n sufficiently large, there exist graphs of order n , not containing a complete graph on s vertices, in which every relatively not too small subset of vertices spans a complete graph on r vertices. Our results improve on previous results of Bollobas and Hind.

Abstract: We prove that almost every r-regular digraph is Hamiltonian for all fixed r ≥ 3.

Abstract: The Erdős-Turan law gives a normal approximation for the order of a randomly chosen permutation of n objects. In this paper, we provide a sharp error estimate for the approximation, showing that, if the mean of the approximating normal distribution is slightly adjusted, the error is of order log−1/2n.

Abstract: We show that the random insertion method for the traveling salesman problem (TSP) may produce a tour (log log n= log log log n) times longer than the optimal tour. The lower bound holds even in the Euclidean Plane. This is in contrast to the fact that the random insertion method performs extremely well in practice. In passing we show that other insertion methods may produce tours (log n= log log n) times longer than the optimal one. No non-constant lower bounds were previously known.

Abstract: We introduce five probability models for random topological graph theory. For two of these models (I and II), the sample space consists of all labeled orientable 2-cell imbeddings of a fixed connected graph, and the interest centers upon the genus random variable. Exact results are presented for the expected value of this random variable for small-order complete graphs, for closed-end ladders, and for cobblestone paths. The expected genus of the complete graph is asymptotic to the maximum genus. For Model III, the sample space consists of all labeled 2-cell imbeddings (possibly nonorientable) of a fixed connected graph, and for Model IV the sample space consists of all such imbeddings with a rotation scheme also fixed. The event of interest is that the ambient surface is orientable. In both these models the complete graph is almost never orientably imbedded. The probability distribution in Models I and III is uniform; in Models II and IV it depends on a parameter p and is uniform precisely when p = 1/2. Model V combines the features of Models II and IV.

Abstract: A graph G is threshold if there exists a ‘weight’ function w: V(G) → R such that the total weight of any stable set of G is less than the total weight of any non-stable set of G. Let n denote the set of threshold graphs with n vertices. A graph is called n-universal if it contains every threshold graph with n vertices as an induced subgraph. n-universal threshold graphs are of special interest, since they are precisely those n-universal graphs that do not contain any non-threshold induced subgraph.In this paper we shall study minimumn-universal (threshold) graphs, i.e.n-universal (threshold) graphs having the minimum number of vertices.It is shown that for any n ≥ 3 there exist minimum n-universal graphs, which are themselves threshold, and others which are not.Two extremal minimum n-universal graphs having respectively the minimum and the maximum number of edges are described, it is proved that they are unique, and that they are threshold graphs.The set of all minimum n-universal threshold graphs is then described constructively; it is shown that it forms a lattice isomorphic to the n − 1 dimensional Boolean cube, and that the minimum and the maximum elements of this lattice are the two extremal graphs introduced above.The proofs provide a (polynomial) recursive procedure that determines for any threshold graph G with n vertices and for any minimum n-universal threshold graph T, an induced subgraph G' of T isomorphic to G.

Abstract: An [n, k, r]-hypergraph is a hypergraph :Yf = (V, E) whose vertex set V is partitioned into n k-element sets V1, V2, ... , Vn and for which, for each choice of r indices, 1 :::;; i1 < i2 < ... < ir :::;; n, there is exactly one edge e E E such that len Vii = 1 if i E {i1, i2, ... , ir} and otherwise le n Vii = 0. An independent transversal of an [n, k, r ]-hypergraph is a set T = {a1,a2, .. . ,an}~ V such that ai E Vi fori= 1, 2, ... ,nand e 1:. T for all e E E. The purpose of this note is to estimate fr(k), defined as the largest n for which any [n,k,r]hypergraph has an independent transversal. The sharpest results are for r = 2 and for the case when k is small compared to r.

Abstract: We define and efficiently compute the canonical flow on a graph, which is a certain feasible solution for the concurrent flow problem and exhibits invariance under the action of the automorphism group of the graph. Using estimates for the congestion of our canonical flow, we derive lower bounds on the crossing number, bisection width, and the edge and vertex expansion of a graph in terms of sizes of the edge and vertex orbits and the average distance in the graph. We further exhibit classes of graphs for which our lower bounds are tight within a multiplicative constant. Also, in cartesian product graphs a concurrent flow is constructed in terms of the concurrent flows in the factors, and in this way lower bounds for the edge and vertex expansion of the power graphs are derived in terms of that of the original graph.

Abstract: Let f be a de Morgan read-once function of n variables. Let f e be the random restriction obtained by independently assigning to each variable of f , the value 0 with probability (1 -e)/2, the value 1 with the same probability, and leaving it unassigned with probability e. We show that f e depends, on the average, on only O (e α n + e n 1/α ) variables, where . This result is asymptotically the tightest possible. It improves a similar result obtained recently by Hastad, Razborov and Yao.

Abstract: Paul Erdős has conjectured that Menger's theorem extends to infinite graphs in the following way: whenever A, B are two sets of vertices in an infinite graph, there exist a set of disjoint A−B paths and an A−B separator in this graph such that the separator consists of a choice of precisely one vertex from each of the paths. We prove this conjecture for graphs that contain a set of disjoint paths to B from all but countably many vertices of A. In particular, the conjecture is true when A is countable.

Abstract: We show that, for any finite set P of points in the plane and for any integer k ≥ 2, there is a finite set R = R(P, k) with the following property: for any k-colouring of R there is a monochromatic set , ⊆ R, such that is combinatorially equivalent to the set P, and the convex hull of P contains no point of R \ . We also consider related questions for colourings of p-element subsets of R (p > 1), and show that these analogues have negative solutions.

Abstract: Whereas the cylindrical version of an Eden cluster in the plane has a surface roughness with a fractal dimension predicted by theory, the central version has hitherto seemed to conflict with theory. However, a fresh way of analysing computer simulations of the central version shows that this anomaly is more apparent than real, and the central version can thereby be reconciled with theory. As a by-product, we obtain statistical data on the properties of the central version in the plane. The macroscopic shape of a central cluster is not circular, and microscopic roughness depends weakly upon the angular direction of portions of the surface. Rather surprisingly, the edge method of construction gives a more nearly circular shape than the external and internal methods. For higher dimensions than the plane, the corresponding treatment is more difficult, and there the situation remains obscure. Higher dimensions and certain other clusters (e.g. Richardson clusters) are treated briefly in Section 6. The theory of surface roughness uses a spatial generalization of martingales, called a serial harness: this is also described in Section 6.

Abstract: Gallai [1] raised the question of determining t(n), the maximum number of triangles in graphs of n vertices with acyclic neighborhoods. Here we disprove his conjecture (t(n) ~ n2/8) by exhibiting graphs having n2/7.5 triangles. We improve the upper bound [11] of (n2 − n)/6 to t(n) ≤; n2/7.02 + O(n). For regular graphs, we further decrease this bound to n2/7.75 + O(n).

Abstract: In [1] we introduced and studied for product hypergraphs where ℋi = (i,ℰi), the minimal size π(ℋn) of a partition of into sets that are elements of . The main result was thatif the ℋis are graphs with all loops included. A key step in the proof concerns the special case of complete graphs. Here we show that (1) also holds when the ℋi are complete d-uniform hypergraphs with all loops included, subject to a condition on the sizes of the i. We also present an upper bound on packing numbers.

Abstract: A Steiner quadruple system SQS(v) of order v is a family ℬ of 4-element subsets of a v-element set V such that each 3-element subset of V is contained in precisely one B ∈ ℬ. We prove that if T ∩ B ≠ o for all B ∈ ℬ (i.e., if T is a transversal), then |T| ≥ v/2, and if T is a transversal of cardinality exactly v/2, then V \ T is a transversal as well (i.e., T is a blocking set). Also, in respect of the so-called ‘doubling construction’ that produces SQS(2v) from two copies of SQS(v), we give a necessary and sufficient condition for this operation to yield a Steiner quadruple system with blocking sets.

Abstract: An element e of a matroid M is called non-binary when M \ e and M/e are both non-binary matroids. Oxley in [5] gave a characterization of the 3-connected non-binary matroids without non-binary elements. In this paper, we will construct all the 3-connected matroids having 1, 2 or 3 non-binary elements.

Abstract: We give some sufficient conditions for an ( S, U )-outline T -factorization of K n to be an ( S, U )-amalgamated T -factorization of K n . We then apply these to give various necessary and sufficient conditions for edge coloured graphs G to have recoverable embeddings in T -factorized K n 's.

Abstract: Let ℝ+(ℋn),ℤ(ℋn),ℤ+(ℋn) be, respectively, the cone over ℝ, the lattice and the cone over ℤ, generated by all cuts of the complete graph on n nodes. For i ≥ 0, let has exactly i realizations in ℤ+(ℋn)}. We show that is infinite, except for the undecided case and empty and for i = 0, n ≤ 5 and for i ≥ 2, n ≤ 3. The set contains 0,1,∞ nonsimplicial points for n ≤ 4, n = 5, n ≥ 6, respectively. On the other hand, there exists a finite number t(n) such that t(n)d ∈ ℤ+(ℋn) for any ; we also estimate such scales for classes of points. We construct families of points of and ℤ+(ℋn), especially on a 0-lifting of a simplicial facet, and points d ∈ ℝ+(ℋn) with di, n = t for 1 ≤ i ≤ n − 1.

Abstract: Let h(·) be an arrangement increasing function, let X have an arrangement increasing density, and let XE be a random permutation of the coordinates of X. We prove E{h(XE)} ≤ E{h(X)}. This comparison is delicate in that similar results are sometimes true and sometimes false. In a finite distributive lattice, a similar comparison follows from Holley's inequality, but the set of permutations with the arrangement order is not a lattice. On the other hand, the set of permutations is a lattice, though not a distributive lattice, if it is endowed with a different partial order, but in this case the comparison does not hold.

Abstract: All digraphs are determined that have the property that when any vertex and any edge that are not adjacent are deleted, the connectivity number decreases by two.

Abstract: We consider the oriented binary tree and the oriented hypercube. The tree edges are oriented from the root to the leaves, while the orientation of the cube edges is induced by the direction from 0 to 1 in the coordinatewise form. The problem is to embed such a tree with l levels into the oriented n-cube as an oriented subgraph, for minimal possible n. A new approach to such problems is presented, which improves the known upper bound n/l ≤ 3/2 given by Havel [1] to n/l ≤ 4/3 + o(1) as l → ∞.

Abstract: In 11] it is shown that the theory of almost all graphs is rst order complete. Furthermore , in 3] a collection of rst order axioms are given from which any rst order property or its negation can be deduced. Here we show that almost all Steinhaus graphs satisfy the axioms of almost all graphs and conclude that a rst order property is true for almost all graphs if and only if it is true for almost all Steinhaus graphs. We also show that certain classes of subgraphs of vertex transitive graphs are rst order complete. Finally, we give a new class of higher order axioms from which it follows that large subgraphs of speciied type exist in almost all graphs.

Abstract: An infinite graph is called bounded if for every labelling of its vertices with natural numbers there exists a sequence of natural numbers which eventually exceeds the labelling along any ray in the graph. Thomassen has conjectured that a countable graph is bounded if and only if its edges can be oriented, possibly both ways, so that every vertex has finite out-degree and every ray has a forward oriented tail. We present a counterexample to this conjecture.

Abstract: In group testing, sets of data undergo tests that reveal if a set contains faulty data. Assuming that data items are faulty with given probability and independently of one another, we investigate small families of tests that enable us to locate correctly all faulty data with probability converging to one as the amount of data grows. Upper and lower bounds on the minimum number of such tests are established for different probability functions, and respective location strategies are constructed.