Tutte proved that if e is an element of a 3-connected matroid M such that neither M\e nor M/e is 3-connected, then e is in a 3-circuit or a 3-cocircuit. In this paper, we prove a broad generalization of this result. Among the consequences of this generalization are that if X is an (n − 1)-element subset of an n-connected matroid M such that neither M\X nor M/X is connected, then, provided |E(M)| ≥ 2(n − 1)≥ 4, X is in both an n-element circuit and an n-element cocircuit. When n = 3, we describe the structure of M more closely using Δ − Y exchanges. Several related results are proved and we also show that, for all fields F other than GF(2), the set of excluded minors for F-representability is closed under both Δ − Y and Y − Δ exchanges.

If particles are dropped randomly on a lattice, with a placement being cancelled if the site in question or a nearest neighbor is already occupied, an ensemble of restricted random walks is created. We seek the time dependence of the expected occupation of a given site. It is shown that this problem reduces to one of enumerating walks from the given site in which a move can only be made to a previously occupied site or one of its nearest neighbors.

The conditional independences within a system of four discrete random variables are studied simultaneously. The problem of where independences can occur at the same time, called the problem of probabilistic representability, is attacked by an analysis of cones of polymatroids. New results on the cone of all polymatroids satisfying Ingleton inequalities imply a substantial reduction of the problem and an explicit description of the remaining open cases.†

To partition the edges of a chordal graph on n vertices into cliques may require as many as n2/6 cliques; there is an example requiring this many, which is also a threshold graph and a split graph. It is unknown whether this many cliques will always suffice. We are able to show that (1 − c)n2/4 cliques will suffice for some c > 0.

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.

Suppose that a process begins with n isolated vertices, to which edges are added randomly one by one so that the maximum degree of the induced graph is always bounded above by d. We prove that if n → ∞ with d fixed, then with probability tending to 1, the final result of this process is a graph with ⌊nd / 2⌋ edges.

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.

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].

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).

Numerous new properties of stochastic conditional independence are introduced. They are aimed, together with two surprisingly trivial examples, at a further reduction of the problem of probabilistic representability for four-element sets, i.e. of the problem which conditional independences within a system of four random variables can occur simultaneously. Proofs are based on fundamental properties of conditional independence and, in the discrete case, on the use of I-divergence and algebraic manipulations with marginal probabilities. A duality question is answered in the negative.

In [11] it is shown that the theory of almost all graphs is first-order complete. Furthermore, in [3] a collection of first-order axioms are given from which any first-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 first-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 first-order complete. Finally, we give a new class of higher-order axioms from which it follows that large subgraphs of specified type exist in almost all graphs.

Along different curves and at different points of the (x, y)-plane the Tutte polynomial evaluates a wide range of quantities. Some of these, such as the number of spanning trees of a graph and the partition function of the planar Ising model, can be computed in polynomial time, others are # P-hard. Here we give a complete characterisation of which points and curves are easy/hard in the bipartite case.

Let ℱn be the set of random mappings ϕ : {1,…,n} → {1,…,n} (such that every mapping is equally likely). For x ε {l,…,n} the elements are called the predecessors of x. Let Nr denote the random variable which counts the number of points x ε {l,…,n} with exactly r predecessors. In this paper we identify the limiting distribution of Nr as n → ∞. If r = r(n) = o(n⅔) then the limiting distribution is Gaussian, if r ˜ Cn⅔ then it is Poisson, and in the remaining case rn−⅔ → ∞ it is degenerate. Furthermore, it is shown that Nr is a Poisson approximation if r → ∞.

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 → ∞.

It is shown that unless NP collapses to random polynomial time RP, there can be no fully polynomial randomised approximation scheme for the antiferromagnetic version of the Q-state Potts model.

The Erdős-Turán 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.

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.