A graph G is called n-minimizable if it can be reduced, by deleting a set of its edges, to a minimally n-connected graph. It is shown that, if n-connected graphs G and H differ only by finitely many vertices and edges, then G is n-minimizable if and only if H is n-minimizable (Theorem 4.12). In the main result, conditions are given that a tree decomposition of an n-connected graph G must satisfy in order to guarantee that the n-minimizability of each of the members of this decomposition implies the n-minimizability of the graph G (Theorem 6.5).

We consider networks in which both the nodes and the links may fail. We represent the network by an undirected graph G. Vertices of the graph fail with probability p and edges of the graph fail with probability q, where all failures are assumed independent. We shall be concerned with minimising the probability P(G) that G is disconnected for graphs with given numbers of vertices and edges. We show how to construct these optimal graphs in many cases when p and q are ‘small’.

For T ∈ GLn (Fq), let Ωn (t, T) be the number of irreducible factors of degree less than or equal to nt in the characteristic polynomial of T. Let

and suppose T is chosen from G Ln(Fq) at random uniformly. We prove that the stochastic process ≺Zn(t)≻t∈[0, 1] converges to the standard Brownian motion process W(t), as n → ∞.

We describe a time randomized algorithm that estimates the number of feasible solutions of a multidimensional knapsack problem within 1 ± ε of the exact number. (Here r is the number of constraints and n is the number of integer variables.) The algorithm uses a Markov chain to generate an almost uniform random solution to the problem.

We consider the set of polynomials of degree n over a finite field and put the uniform probability measure on this set. Any such polynomial factors uniquely into a product of its irreducible factors. To each polynomial we associate a step function on the interval [0,1] such that the size of each jump corresponds to the number of factors of a certain degree in the factorization of the random polynomial. We normalize these random functions and show that the resulting random process converges weakly to Brownian motion as n → ∞. This result complements earlier work by the author on the order statistics of the degree sequence of the factors of a random polynomial.

Many of the classical results of Ramsey Theory, including those of Hilbert, Schur, and van der Waerden, are naturally stated as instances of the following problem: given a u × ν matrix A with rational entries, is it true, that whenever the set ℕ of positive integers is finitely coloured, there must exist some x∈ℕν such that all entries of Ax are the same colour? While the theorems cited are all consequences of Rado's theorem, the general problem had remained open. We provide here several solutions for the alternate problem, which asks that x∈ℕν. Based on this, we solve the general problem, giving various equivalent characterizations.

A theorem of Makarov states that the harmonic measure of a connected subset of ℝ2 is supported on a set of Hausdorff dimension one. This paper gives an analogue of this theorem for discrete harmonic measure, i.e., the hitting measure of simple random walk. It is proved that for any 1/2 < α < 1, β < α − 1/2, there is a constant k such that for any connected subset A ⊂ ℤ2 of radius n,

where HA denotes discrete harmonic measure.

In this paper we consider the problem of estimating the spectral gap of a reversible Markov chain in terms of geometric quantities associated with the underlying graph. This quantity provides a bound on the rate of convergence of a Markov chain towards its stationary distribution. We give a critical and systematic treatment of this subject, summarizing and comparing the results of the two main approaches in the literature, algebraic and functional. The usefulness and drawbacks of these bounds are also discussed here.

In this paper we consider the following problem, given a graph H, what is the structure of a typical, i.e. random, H-free graph? We completely solve this problem for all graphs H containing a critical vertex. While this result subsumes a sequence of known results, its short proof is self contained.

Let G be a connected graph with n vertices and m edges (multiple edges allowed), and let k ≥ 2 be an integer. There is an algorithm with (optimal) running time of O(m) that finds

(i) a bipartite subgraph of G with ≥ m/2 + (n − 1)/4 edges,

(ii) a bipartite subgraph of G with ≥ m/2 + 3(n−1)/8 edges if G is triangle-free,

(iii) a k-colourable subgraph of G with ≥ m − m/k + (n−1)/k + (k − 3)/2 edges if k ≥ 3 and G is not k-colorable.

Infinite families of graphs show that each of those lower bounds on the worst-case performance are best possible (for every algorithm). Moreover, even if short cycles are excluded, the general lower bound of m − m/k cannot be replaced by m − m/k + εm for any fixed ε > 0; and it is NP-complete to decide whether a graph with m edges contains a k-colorable subgraph with more than m − m/k + εm edges, for any k ≥ 2 and ε> 0, ε < 1/k.

Let G be a graph and P(G, t) be the chromatic polynomial of G. It is known that P(G, t) has no zeros in the intervals (−∞, 0) and (0, 1). We shall show that P(G, t) has no zeros in (1, 32/27]. In addition, we shall construct graphs whose chromatic polynomials have zeros arbitrarily close to 32/27.

The main result of this paper gives a structure of a triangle-free graph of order n with minimal degree greater than 10n/29. This extends results given by Andrásfai et al. [1] and Häggkvist [6].

This is a survey of a number of recent papers dealing with graphs from a geometric perspective. The main theme of these studies is the relationship between graph properties that are local in nature, and global graph parameters. Connections with the theory of distributed computing are pointed out and many open problems are presented.

A transformation that increases the blocking probability of the channel graphs arising in interconnection networks is developed. This provides the basis for eminently computing a bound on the blocking probability by applying the transformation to reduce an arbitrary four-stage channel graph to a threshold channel graph. The four-stage channel graph that is most likely to block, given links that are equally likely to block, is characterized. Partial results are proved concerning the threshold channel graphs that are least likely to block.

We prove that the probability i(n, k) that a random permutation of an n element set has an invariant subset of precisely k elements decreases as a power of k, for k ≤ n/2. Using this fact, we prove that the fraction of elements of Sn belong to transitive subgroups other than Sn or An tends to 0 when n → ∞, as conjectured by Cameron. Finally, we show that for every ∈ > 0 there exists a constant C such that C elements of the symmetric group Sn, chosen randomly and independently, generate invariably Sn with probability at least 1 − ∈. This confirms a conjecture of McKay.

A new graph theoretic proof of the convergence of Markov chains with variable transition probabilities and a new algorithm for computing the limiting distributions are presented.