We analyse the weighted height of random tries built from independent strings of i.i.d. symbols on the finite alphabet {1, . . .d}. The edges receive random weights whose distribution depends upon the number of strings that visit that edge. Such a model covers the hybrid tries of de la Briandais and the TST of Bentley and Sedgewick, where the search time for a string can be decomposed as a sum of processing times for each symbol in the string. Our weighted trie model also permits one to study maximal path imbalance. In all cases, the weighted height is shown to be asymptotic to c log n in probability, where c is determined by the behaviour of the core of the trie (the part where all nodes have a full set of children) and the fringe of the trie (the part of the trie where nodes have only one child and form spaghetti-like trees). It can be found by maximizing a function that is related to the Cramér exponent of the distribution of the edge weights.

We formulate and give partial answers to several combinatorial problems on volumes of simplices determined by n points in 3-space, and in general in d dimensions.

1. (i) The number of tetrahedra of minimum (non-zero) volume spanned by n points in 3 is at most , and there are point sets for which this number is . We also present an O(n3) time algorithm for reporting all tetrahedra of minimum non-zero volume, and thereby extend an algorithm of Edelsbrunner, O'Rourke and Seidel. In general, for every , the maximum number of k-dimensional simplices of minimum (non-zero) volume spanned by n points in d is Θ(nk).

2. (ii) The number of unit volume tetrahedra determined by n points in 3 is O(n7/2), and there are point sets for which this number is Ω(n3 log logn).

3. (iii) For every , the minimum number of distinct volumes of all full-dimensional simplices determined by n points in d, not all on a hyperplane, is Θ(n).

We prove that the chromatic polynomial of a finite graph of maximal degree Δ is free of zeros for |q| ≥ C*(Δ) with This improves results by Sokal and Borgs. Furthermore, we present a strengthening of this condition for graphs with no triangle-free vertices.

Semi-graphoids are combinatorial structures that arise in statistical learning theory. They are equivalent to convex rank tests and to polyhedral fans that coarsen the reflection arrangement of the symmetric group Sn. In this paper we resolve two problems on semi-graphoids posed in Studený's book (2005), and we answer a related question of Postnikov, Reiner and Williams on generalized permutohedra. We also study the semigroup and the toric ideal associated with semi-graphoids.

We consider the number of vertices that must be removed from a graph G in order that the remaining subgraph have no component with more than k vertices. Our principal observation is that, if G is a sparse random graph or a random regular graph on n vertices with n → ∞, then the number in question is essentially the same for all values of k that satisfy both k → ∞ and k =o(n).

A proper vertex colouring of a graph is equitable if the sizes of colour classes differ by at most one. We present a new shorter proof of the celebrated Hajnal–Szemerédi theorem: for every positive integer r, every graph with maximum degree at most r has an equitable colouring with r+1 colours. The proof yields a polynomial time algorithm for such colourings.

Let G be a graph with n vertices, and let k be an integer dividing n. G is said to be strongly k-colourable if, for every partition of V(G) into disjoint sets V1 ∪ ··· ∪ Vr, all of size exactly k, there exists a proper vertex k-colouring of G with each colour appearing exactly once in each Vi. In the case when k does not divide n, G is defined to be strongly k-colourable if the graph obtained by adding isolated vertices is strongly k-colourable. The strong chromatic number of G is the minimum k for which G is strongly k-colourable. In this paper, we study the behaviour of this parameter for the random graph Gn,p. In the dense case when p ≫ n−1/3, we prove that the strong chromatic number is a.s. concentrated on one value Δ + 1, where Δ is the maximum degree of the graph. We also obtain several weaker results for sparse random graphs.

We analyse classes of planar graphs with respect to various properties such as polynomial-time solvability of the dominating set problem or boundedness of the tree-width. A helpful tool to address this question is the notion of boundary classes. The main result of the paper is that for many important properties there are exactly two boundary classes of planar graphs.

A famous result of Freĭman describes the sets A, of integers, for which |A+A| ≤ K|A|. In this short note we address the analogous question for subsets of vector spaces over . Specifically we show that if A is a subset of a vector space over with |A+A| ≤ K|A| then A is contained in a coset of size at most 2O(K3/2 log K)|A|, which improves upon the previous best, due to Green and Ruzsa, of 2O(K2)|A|. A simple example shows that the size may need to be at least 2Ω(K)|A|.

We prove tail estimates for variables of the form ∑if(Xi), where (Xi)i is a sequence of states drawn from a reversible Markov chain, or, equivalently, from a random walk on an undirected graph. The estimates are in terms of the range of the function f, its variance, and the spectrum of the graph. The purpose of our estimates is to determine the number of chain/walk samples which are required for approximating the expectation of a distribution on vertices of a graph, especially an expander. The estimates must therefore provide information for fixed number of samples (as in Gillman's [4]) rather than just asymptotic information. Our proofs are more elementary than other proofs in the literature, and our results are sharper. We obtain Bernstein- and Bennett-type inequalities, as well as an inequality for sub-Gaussian variables.