Abstract: A question recently posed by Haggkvist and Scott asked whether or not there exists a constant c such that, if G is a graph of minimum degree ck, then G contains cycles of k consecutive even lengths. In this paper we answer the question by proving that, for k > 2, a bipartite graph of average degree at least 4k and girth g contains cycles of (g/2 − 1)k consecutive even lengths. We also obtain a short proof of the theorem of Bondy and Simonovits, that a graph of order n and size at least 8(k − 1)n1+1/k has a cycle of length 2k.

Abstract: Let Gp be a random graph on 2d vertices where edges are selected independently with a fixed probability p > ¼, and let H be the d-dimensional hypercube Qd. We answer a question of Bollobas by showing that, as d → ∞, Gp almost surely has a spanning subgraph isomorphic to H. In fact we prove a stronger result which implies that the number of d-cubes in G ∈ G(n, M) is asymptotically normally distributed for M in a certain range. The result proved can be applied to many other graphs, also improving previous results for the lattice, that is, the 2-dimensional square grid. The proof uses the second moment method – writing X for the number of subgraphs of G isomorphic to H, where G is a suitable random graph, we expand the variance of X as a sum over all subgraphs of H itself. As the subgraphs of H may be quite complicated, most of the work is in estimating the various terms of this sum.

Abstract: Two results dealing with the relation between the smallest eigenvalue of a graph and its bipartite subgraphs are obtained. The first result is that the smallest eigenvalue μ of any non-bipartite graph on n vertices with diameter D and maximum degree Δ satisfies μ ≥ −Δ + 1/(D+1)n. This improves previous estimates and is tight up to a constant factor. The second result is the determination of the precise approximation guarantee of the MAX CUT algorithm of Goemans and Williamson for graphs G = (V, E) in which the size of the max cut is at least A∣E∣, for all A between 0.845 and 1. This extends a result of Karloff.

Abstract: In this paper, we combine two previous works, the first being by the first author and K. Nelander, and the second by J. van den Berg and the second author, to show (1) that one can carry out a Propp–Wilson exact simulation for all Markov random fields on Zd satisfying a certain high noise assumption, and (2) that all such random fields are a finitary image of a finite state i.i.d. process. (2) is a strengthening of the previously known fact that such random fields are so-called Bernoulli shifts.

Abstract: Let 0 < p < 1, q = 1 − p and b be fixed and let G ∈ G(n, p) be a graph on n vertices where each pair of vertices is joined independently with probability p. We show that the probability that every vertex of G has degree at most pn + b √npq is equal to (c + o(1))n, for c = c(b) the root of a certain equation. Surprisingly, c(0) = 0.6102 … is greater than ½ and c(b) is independent of p. To obtain these results we consider the complete graph on n vertices with weights on the edges. Taking these weights as independent normal N(p, pq) random variables gives a ‘continuous’ approximation to G(n, p) whose degrees are much easier to analyse.

Abstract: The intersection exponent ξ for simple random walk in two and three dimensions gives a measure of the rate of decay of the probability that paths do not intersect. In this paper we show that the intersection exponent for random walks is the same as that for Brownian motion and show in fact that the probability of nonintersection up to distance n is comparable (equal up to multiplicative constants) to n−ξ.

Abstract: Let F = {G1, …, Gt} be a family of n-vertex graphs defined on the same vertex-set V, and let k be a positive integer. A subset of vertices D ⊂ V is called an (F, k)-core if, for each v ∈ V and for each i = 1, …, t, there are at least k neighbours of v in Gi that belong to D. The subset D is called a connected (F, k)-core if the subgraph induced by D in each Gi is connected. Let δi be the minimum degree of Gi and let δ(F) = minti=1δi. Clearly, an (F, k)-core exists if and only if δ(F) ≥ k, and a connected (F, k)-core exists if and only if δ(F) ≥ k and each Gi is connected. Let c(k, F) and cc(k, F) be the minimum size of an (F, k)-core and a connected (F, k)-core, respectively. The following asymptotic results are proved for every t < ln ln δ and k < √lnδ:formula hereThe results are asymptotically tight for infinitely many families F. The results unify and extend related results on dominating sets, strong dominating sets and connected dominating sets.

Abstract: Consider the integer lattice L = Z2. For some m ≥ 4, let us colour each column of this lattice independently and uniformly with one of m colours. We do the same for the rows, independently of the columns. A point of L will be called blocked if its row and column have the same colour. We say that this random configuration percolates if there is a path in L starting at the origin, consisting of rightward and upward unit steps, avoiding the blocked points. As a problem arising in distributed computing, it has been conjectured that for m ≥ 4 the configuration percolates with positive probability. This question remains open, but we prove that the probability that there is percolation to distance n but not to infinity is not exponentially small in n. This narrows the range of methods available for proving the conjecture.

Abstract: Given a graph G = (V, E) and a set of κ pairs of vertices in V, we are interested in finding, for each pair (ai, bi), a path connecting ai to bi such that the set of κ paths so found is edge-disjoint. (For arbitrary graphs the problem is NP-complete, although it is in P if κ is fixed.)We present a polynomial time randomized algorithm for finding edge-disjoint paths in the random regular graph Gn,r, for sufficiently large r. (The graph is chosen first, then an adversary chooses the pairs of end-points.) We show that almost every Gn,r is such that all sets of κ = Ω(n/log n) pairs of vertices can be joined. This is within a constant factor of the optimum.

Abstract: Let A be an m × n matrix and q := ⌊log2(m)⌋+1. In this article we improve the well-known bound lindisc(A) ≤ 2 herdisc(A) and show thatformula hereAs with the previous proofs relating to this problem, ours is constructive. We will give an on-line algorithm and analyse it using game theory.

Abstract: This paper describes methods for counting the number of nonnegative integer solutions of the system Ax = b when A is a nonnegative totally unimodular matrix and b an integral vector of fixed dimension. The complexity (under a unit cost arithmetic model) is strong in the sense that it depends only on the dimensions of A and not on the size of the entries of b. For the special case of ‘contingency tables’ the run-time is 2O(√dlogd) (where d is the dimension of the polytope). The method is complementary to Barvinok's in that our algorithm is effective on problems of high dimension with a fixed number of (non-sign) constraints, whereas Barvinok's algorithms are effective on problems of low dimension and an arbitrary number of constraints.

Abstract: We prove that if the edge probability p(n) satisfies n−1/4+e ≤ p(n) ≤ 3/4, where 0 < e < 1/4 is a constant, then the choice number and the chromatic number of the random graph G(n, p) are almost surely asymptotically equal.

Abstract: An asymptotics, as n → ∞, for the expected number of distinct part sizes in a random composition of an integer n is obtained.

Abstract: Let G be an abelian group. For a subset A ⊂ G, denote by 2 ∧ A the set of sums of two different elements of A. A conjecture by Erdos and Heilbronn, first proved by Dias da Silva and Hamidoune, states that, when G has prime order, ∣2 ∧ A∣ ≥ min(∣G∣, 2∣A∣ − 3).We prove that, for abelian groups of odd order (respectively, cyclic groups), the inequality ∣2 ∧ A∣ ≥ min(∣G∣, 3∣A∣/2) holds when A is a generating set of G, 0 ∈ A and ∣A∣ ≥ 21 (respectively, ∣A∣ ≥ 33). The structure of the sets for which equality holds is also determined.

Abstract: The maximum expected length of an increasing subsequence which can be selected by a non-anticipating policy from a random permutation of 1, …, n is known to be asymptotic to √2n. We give a new proof of this fact and demonstrate a policy which achieves this value.

Abstract: Consider sequences {Xi}mi=1 and {Yj}nj=1 of independent random variables, taking values in a finite alphabet, and assume that the variables X1, X2, … and Y1, Y2, … follow the distributions μ and v, respectively. Two variables Xi and Yj are said to match if Xi = Yj. Let the number of matching subsequences of length k between the two sequences, when r, 0 ≤ r < k, mismatches are allowed, be denoted by W.In this paper we use Stein's method to bound the total variation distance between the distribution of W and a suitably chosen compound Poisson distribution. To derive rates of convergence, the case where E[W] stays bounded away from infinity, and the case where E[W] → ∞ as m, n → ∞, have to be treated separately. Under the assumption that ln n/ln(mn) → ρ ∈ (0, 1), we give conditions on the rate at which k → ∞, and on the distributions μ and v, for which the variation distance tends to zero.

Abstract: A type of evolution of graphs with maximum vertex degree at most d is introduced. This evolution can start from any initial graph whose set of vertices of degree less than d is independent. The main concern is the regularity of graphs generated by this graph process when the initial graph has no edges. By analysis of the solutions of systems of differential equations it is shown that the final graph of this evolution is asymptotically almost surely a d-regular graph (subject to the usual parity condition).

Abstract: We consider graphs G n generated by multisets [Iscr ] n with n random integers as elements, such that vertices of G n are connected by edges if the elements of [Iscr ] n that the vertices represent are the same, and prove asymptotic results on the sparsity of edges connecting the different subgraphs G n of the random graph generated by ∪ ∞ n =1 [Iscr ] n . These results are of independent interest and, for two models of the bootstrap, we also use them here to link almost sure and complete convergence of the corresponding bootstrap means and averages of related randomly chosen subsequences of a sequence of independent and identically distributed random variables with a finite mean. Complete convergence of these means and averages is then characterized in terms of a relationship between a moment condition on the bootstrapped sequence and the bootstrap sample size. While we also obtain new sufficient conditions for the almost sure convergence of bootstrap means, the approach taken here yields the first necessary conditions.

Abstract: For a subgroup W of the hyperoctahedral group On which is generated by reflections, we consider the linear dependence matroid MW on the column vectors corresponding to the reflections in W. We determine all possible automorphism groups of MW and determine when W ≅ = Aut(MW). This allows us to connect combinatorial and geometric symmetry. Applications to zonotopes are also considered. Signed graphs are used as a tool for constructing the automorphisms.

Abstract: For any fixed l < k we present families of asymptotically good packings and coverings of the l-subsets of an n-element set by k-subsets, and an algorithm that, given a natural number i, finds the ith k-subset of the family in time and space polynomial in log n.

Abstract: A graph G is quasi 4-connected if it is simple, 3-connected, has at least five vertices, and for every partition (A, B, C) of V(G) either ∣C∣ ≥ 4, or G has an edge with one end in A and the other end in B, or one of A,B has at most one vertex. We show that any quasi 4-connected nonplanar graph with minimum degree at least three and no cycle of length less than five has a minor isomorphic to P−10, the Petersen graph with one edge deleted. We deduce the following weakening of Tutte's Four Flow Conjecture: every 2-edge-connected graph with no minor isomorphic to P−10 has a nowhere-zero 4-flow. This extends a result of Kilakos and Shepherd who proved the same for 3-regular graphs.

Abstract: In [9, p. 469], Oxley made the following conjecture, which is a geometric analogue of a conjecture of Lovasz (see [1, p. 290]) about complete graphs.Conjecture 1.1. Let G be a rank-n GF(q)-representable simple matroid with critical exponent n − γ. If, for every coline X in G, c(G/X; q) = c(G; q) − 2 = n − γ − 2, then G is the projective geometry PG(n − 1, q).We shall call the rank n, the critical ‘co-exponent’ γ, and the order q of the field the parameters of Oxley's conjecture. We exhibit several counterexamples to this conjecture. These examples show that, for a given prime power q and a given positive integer γ, Oxley's conjecture holds for only finitely many ranks n. We shall assume familiarity with matroid theory and, in particular, the theory of critical problems. See [6] and [9].A subset C of points of PG(n − 1, q) is a (γ, k)-cordon if, for every k-codimensional subspace X in PG(n − 1, q), the intersection C ∩ X contains a γ-dimensional subspace of PG(n − 1, q). In this paper, our primary interest will be in constructing (γ, 2)-cordons. With straightforward modifications, our methods will also yield (γ, k)-cordons.Complements of counterexamples to Oxley's conjecture are (γ, 2)-cordons.

Abstract: The k-colouring problem is to colour a given k-colourable graph with k colours. This problem is known to be NP-hard even for fixed k ≥ 3. The best known polynomial time approximation algorithms require nδ (for a positive constant δ depending on k) colours to colour an arbitrary k-colourable n-vertex graph. The situation is entirely different if we look at the average performance of an algorithm rather than its worst-case performance. It is well known that a k-colourable graph drawn from certain classes of distributions can be k-coloured almost surely in polynomial time.In this paper, we present further results in this direction. We consider k-colourable graphs drawn from the random model in which each allowed edge is chosen independently with probability p(n) after initially partitioning the vertex set into k colour classes. We present polynomial time algorithms of two different types. The first type of algorithm always runs in polynomial time and succeeds almost surely. Algorithms of this type have been proposed before, but our algorithms have provably exponentially small failure probabilities. The second type of algorithm always succeeds and has polynomial running time on average. Such algorithms are more useful and more difficult to obtain than the first type of algorithms. Our algorithms work as long as p(n) ≥ n−1+e where e is a constant greater than 1/4.

Abstract: Let G = (V, E) be a graph with n vertices, chromatic number χ(G) and list chromatic number χl(G). Suppose each vertex of V(G) is assigned a list of t colours. Albertson, Grossman and Haas [1] conjectured that at least [formula here] vertices can be coloured properly from these lists.Albertson, Grossman and Haas [1] and Chappell [3] proved partial results concerning this conjecture. This paper presents algorithms that colour at least the number of vertices given in the bounds of Albertson, Grossman and Haas, and Chappell. In particular, it follows that the conjecture is valid for all bipartite graphs and that, for every bipartite graph and every assignment of lists with t colours in each list where 0 ≤ t ≤ χl(G), it is possible to colour at least (1 − (1/2)t)n vertices in polynomial time. Thus, if G is bipartite and L is a list assignment with ∣L(v)∣ ≥ log2 n for all v ∈ V, then G is L-list colourable in polynomial time.

Abstract: The core of a graph G is the subgraph GΔ induced by the vertices of maximum degree. We define the deleted core D(G) of G. We extend an earlier sufficient condition due to Hoffman [7] for a graph H to be the core of a Class 2 graph, and thereby provide a stronger sufficient condition. The new sufficient condition is in terms of D(H). We show that in some circumstances our condition is necessary; but it is not necessary in general.

Abstract: The twisted odd graphs are obtained from the well-known odd graphs through an involutive automorphism. As expected, the twisted odd graphs share some of the interesting properties of the odd graphs but, in general, they seem to have a more involved structure. Here we study some of their basic properties, such as their automorphism group, diameter, and spectrum. They turn out to be examples of the so-called boundary graphs, which are graphs satisfying an extremal property that arises from a bound for the diameter of a graph in terms of its distinct eigenvalues.

Abstract: Vapnik and Chervonenkis proposed in [7] a combinatorial notion of dimension that reflects the ‘combinatorial complexity’ of families of sets. In the three decades that have passed since that paper, this notion – the Vapnik–Chervonenkis dimension (VC-dimension) – has been discovered to be of primal importance in quite a wide variety of topics in both pure mathematics and theoretical computer science.In this paper we turn our attention to classes with infinite VC-dimension, a realm thrown into one big bag by the usual VC-dimension analysis. We identify three levels of combinatorial complexity of classes with infinite VC-dimension. We show that these levels fall under the set-theoretic definition of σ-ideals (in particular, each of them is closed under countable unions), and that they are all distinct. The first of these levels (i.e., the family of ‘small’ infinite-dimensional classes) coincides with the family of classes which are non-uniformly PAC-learnable.Maybe the most surprising contribution of this work is the discovery of an intimate relation between the VC-dimension of a class of subsets of the natural numbers and the Lebesgue measure of the set of reals defined when these subsets are viewed as binary representations of real numbers.As a by-product, our investigation of the VC-dimension-induced ideals over the reals yields a new proper extension of the Lebesgue measure. Another offshoot of this work is a simple result in probability theory, showing that, given any sequence of pairwise independent events, any random event is eventually independent of the members of the sequence.

Abstract: We examine the specialization to simple matroids of certain problems in extremal matroid theory that are concerned with bounded cocircuit size. Assume that each cocircuit of a simple matroid M has at most d elements. We show that if M has rank 3, then M has at most d + ⌊√d⌋ + 1 points, and we classify the rank-3 simple matroids M that have exactly d + ⌊√d⌋ points. We show that if M is a connected matroid of rank 4 and d is q3 with q > 1, then M has at most q3 + q2 + q + 1 points; this upper bound is strict unless q is a prime power, in which case the only such matroid with exactly q3 + q2 + q + 1 points is the projective geometry PG(3, q). We also show that if d is q4 for a positive integer q and if M has rank 5 and is vertically 5-connected, then M has at most q4 + q3 + q2 + q + 1 points; this upper bound is strict unless q is a prime power, in which case PG(4, q) is the only such matroid that attains this bound.