The Wiener index is analysed for random recursive trees and random binary search trees in uniform probabilistic models. We obtain expectations, asymptotics for the variances, and limit laws for this parameter. The limit distributions are characterized as the projections of bivariate measures that satisfy certain fixed point equations. Covariances, asymptotic correlations, and bivariate limit laws for the Wiener index and the internal path length are given.

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.

Let Tn be the complete binary tree of height n considered as the Hasse diagram of a poset with its root 1n as the maximum element. Define A(n; T) = [mid ]{S ⊆ Tn : 1n ∈ S, S ≅ T}[mid ], and B(n; T) = [mid ]{S ⊆ Tn : 1n ∉ S, S ≅ T}[mid ]. In this note we prove that for any fixed n and rooted binary trees T1, T2 such that T2 contains a subposet isomorphic to T1. We conjecture that the ratio A/B also increases with T for arbitrary trees. These inequalities imply natural behaviour of the optimal stopping time in a poset extension of the secretary problem.

In this paper we are concerned with the following conjecture.

Conjecture. For any positive integers n and k satisfying k < n, and any sequence a1, a2, … ak of not necessarily distinct elements of Zn, there exists a permutation π ∈ Sk such that the elements aπ(i)+i are all distinct modulo n.

We prove this conjecture when 2k [les ] n+1. We then apply this result to tree embeddings. Specifically, we show that, if T is a tree with n edges and radius r, then T decomposes Kt for some t [les ] 32(2r+4)n2+1.

We give a concentration inequality involving a family of independent random permutations, which is useful for analysing certain randomized methods for graph colouring.

The circuit cover problem for mixed graphs (those containing edges and/or arcs) is defined as follows. Given a mixed graph M with a nonnegative integer weight function p on its edges and arcs, decide whether (M, p) has a circuit cover, that is, a list of circuits in M such that every edge (arc) e is contained in exactly p(e) circuits of the list. In the special case when M is a directed graph (contains only arcs), the problem is easy, but when M is an undirected graph not many results are known. For general mixed graphs this problem was shown to be NP-complete by Arkin and Papadimitriou in 1986. We prove that this problem remains NP-complete for planar mixed graphs. Furthermore, we present a good characterization for the existence of a circuit cover when M is series-parallel (a similar result holds for the fractional version). We also describe a polynomial algorithm to find such a circuit cover, when it exists. This is an ellipsoid-based algorithm whose separation problem is the minimum circuit problem on series-parallel mixed graphs, which we show to be polynomially solvable. Results on two well-known combinatorial problems, the problem of detecting negative circuits and the problem of finding shortest paths, are also presented. We prove that both problems are NP-hard for planar mixed graphs.

Consider the class of graphs on n vertices which have maximum degree at most 1/2n−1+τ, where τ [ges ] −n1/2+ε for sufficiently small ε > 0. We find an asymptotic formula for the number of such graphs and show that their number of edges has a normal distribution whose parameters we determine. We also show that expectations of random variables on the degree sequences of such graphs can often be estimated using a model based on truncated binomial distributions.

We study a model motivated by the minesweeper game. In this model one starts with the percolation of mines on the lattice ℤd, and then tries to find an infinite path of mine-free sites. At every recovery of a free site, the player is given some information on the sites adjacent to the current site. We compare the parameter values for which there exists a strategy such that the process survives to the critical parameter of ordinary percolation. We then derive improved bounds for these values for the same process, when the player has some complexity restrictions in computing his moves. Finally, we discuss some monotonicity issues which arise naturally for this model.

In this paper we study distances in random subgraphs of a generalized n-cube [Qscr ]ns over a finite alphabet S of size s. [Qscr ]ns is the direct product of complete graphs over s vertices, its vertices being the n-tuples (x1, …, xn), with xi ∈ S, i = 1, … n, and two vertices being adjacent if they differ in exactly one coordinate. A random (induced) subgraph γ of [Qscr ]ns is obtained by selecting [Qscr ]ns-vertices with independent probability pn and then inducing the corresponding edges from [Qscr ]ns. Our main result is that dγ (P,Q) [les ] [2k+3]d[Qscr ]ns (P,Q) almost surely for P,Q ∈ γ, pn = n−a and 0 [les ] a < ½, where k = [1+3a/1−2a] and dγ and d[Qscr ]ns denote the distances in γ and [Qscr ]ns, respectively.

Given a family [Fscr ] of r-graphs, let ex(n, [Fscr ]) be the maximum number of edges in an n-vertex r-graph containing no member of [Fscr ]. Let C(r)4 denote the family of r-graphs with distinct edges A, B, C, D, such that A ∩ B = C ∩ D = Ø and A ∪ B = C ∪ D. For s1 [les ] … [les ] sr, let K(r) (s1,…,sr) be the complete r-partite r-graph with parts of sizes s1,…,sr.

Füredi conjectured over 15 years ago that ex(n,C(3)4) [les ] (n2) for sufficiently large n. We prove the weaker result

ex(n, {C(3)4, K(3)(1,2,4)}) [les ] (n2).

Generalizing a well-known conjecture for the Turán number of bipartite graphs, we conjecture that

ex(n, K(r)(s1,…,sr)) = Θ(nr−1/s),

where s = Πr−1i=1si. We prove this conjecture when s1 = … = sr−2 = 1 and

(i) sr−1 = 2, (ii) sr−1 = sr = 3, (iii)sr > (sr−1−1)!.

In cases (i) and (ii), we determine the asymptotic value of ex(n,K(r)(s1,…,sr)).

We also provide an explicit construction giving

ex(n,K(3)(2,2,3)) > (1/6−o(1))n8/3.

This improves upon the previous best lower bound of Ω(n29/11) obtained by probabilistic methods. Several related open problems are also presented.

A top to random shuffle of a deck of cards is performed by taking the top card off of the deck and replacing it in a randomly chosen position of the deck. We find approximations of the relative entropy of a deck of n cards after m successive top to random shuffles. Initially the relative entropy decays linearly and for larger m it decays geometrically at a rate that alters abruptly at m = n log n. It converges to an explicitly given expression when m = [n log n+cn] for a constant c.

On the class of labelled combinatorial structures called assemblies we define complex-valued multiplicative functions and examine their asymptotic mean values. The problem reduces to the investigation of quotients of the Taylor coefficients of exponential generating series having Euler products. Our approach, originating in probabilistic number theory, requires information on the generating functions only in the convergence disc and rather weak smoothness on the circumference. The results could be applied to studying the asymptotic value distribution of decomposable mappings defined on assemblies.

This paper introduces a split-and-merge transformation of interval partitions which combines some features of one model studied by Gnedin and Kerov [12, 11] and another studied by Tsilevich [30, 31] and Mayer-Wolf, Zeitouni and Zerner [21]. The invariance under this split-and-merge transformation of the interval partition generated by a suitable Poisson process yields a simple proof of the recent result of [21] that a Poisson–Dirichlet distribution is invariant for a closely related fragmentation–coagulation process. Uniqueness and convergence to the invariant measure are established for the split-and-merge transformation of interval partitions, but the corresponding problems for the fragmentation–coagulation process remain open.

Let λ(G) be the largest eigenvalue of the adjacency matrix of a graph G: We show that if G is Kp+1-free then

This inequality was first conjectured by Edwards and Elphick in 1983 and supersedes a series of previous results on upper bounds of λ(G).

Let Ti denote the number of all i-cliques of G, λ = λ(G) and p = cl(G): We show

Let δ be the minimal degree of G. We show

This inequality supersedes inequalities of Stanley and Hong. It is sharp for regular graphs and for a class of graphs which are in some sense maximally irregular.

Consider a finite alphabet Ω and patterns which consist of characters from Ω. For a given pattern w, let cor(w) denote its autocorrelation, which can be seen as a measure of the amount of overlap in w. Letting aw(n) denote the number of strings over Ω of length n which do not contain w as a substring, the main result of this paper is: If cor(w) > cor(w′) then aw(n)−aw′(n) > (|Ω|−1)(aw(n−1)−aw′(n−1)) for n [ges ] N, and the value of N is given. This result confirms a conjecture by Eriksson [2], which was previously proved to be true by Cakir, Chryssaphinou and Månsson [1] when |Ω| [ges ] 3.

In ‘Automorphisms of Dowling lattices and related geometries’, J. Bonin constructed the automorphism group A of a Dowling lattice as the image of a certain semidirect product, A = θ(K [rtimes ] H). In this work we find necessary and sufficient conditions for this quotient to be the semidirect product A = θ(K) [rtimes ] θ(H). In addition, we include a construction of A that lends itself to computation more readily than that found in Bonin's work.

Let G be a connected graph that is 2-cell embedded in a surface S, and let G* be its topological dual graph. We will define and discuss several matroids whose element set is E(G), for S homeomorphic to the plane, projective plane, or torus. We will also state and prove old and new results of the type that the dual matroid of G is the matroid of the topological dual G*.

The hexagonal lattice site percolation critical probability is shown to be at most 0.79472, improving the best previous mathematically rigorous upper bound. The bound is derived by using the substitution method to compare the site model with the bond model, the latter of which is exactly solved. Shortcuts which eliminate a substantial amount of computation make the derivation of the bound possible.

We study the complexity of computing the coefficients of three classical polynomials, namely the chromatic, flow and reliability polynomials of a graph. Each of these is a specialization of the Tutte polynomial Σtijxiyj. It is shown that, unless NP = RP, many of the relevant coefficients do not even have good randomized approximation schemes. We consider the quasi-order induced by approximation reducibility and highlight the pivotal position of the coefficient t10 = t01, otherwise known as the beta invariant.

Our nonapproximability results are obtained by showing that various decision problems based on the coefficients are NP-hard. A study of such predicates shows a significant difference between the case of graphs, where, by Robertson–Seymour theory, they are computable in polynomial time, and the case of matrices over finite fields, where they are shown to be NP-hard.

We show that, for every positive integer c*, there is an integer n such that, if M is a matroid whose largest cocircuit has size c*, then E(M) can be partitioned into two sets E1 and E2 such that every connected component of each of M[mid ]E1 and M[mid ]E2 has at most n elements.