Let r = r(n) → ∞ with 3 [les ] r [les ] n1−η for an arbitrarily small constant η > 0, and let Gr denote a graph chosen uniformly at random from the set of r-regular graphs with vertex set {1, 2, …, n}. We prove that, with probability tending to 1 as n → ∞, Gr has the following properties: the independence number of Gr is asymptotically 2n log r/r and the chromatic number of Gr is asymptotically r/2nlogr.

We show that, if G is a graph of order n with maximal degree Δ(G) and minimal degree δ(G) whose complement contains no K2,s, s [ges ] 2, then G contains every tree T of order n−s+1 whose maximal degree is at most Δ(G) and whose vertex of second-largest degree is at most δ(G). We then show that this result implies that special cases of two conjectures are true. We verify that the Erdös–Sós conjecture, which states that a graph whose average degree is larger than k−1 contains every tree of order k+1, is true for graphs whose complement does not contain a K2,4, and the Komlós–Sós conjecture, which states that every graph of median degree at least k contains every tree of order k+1, is true for graphs whose complement does not contain a K2,3.

Let {Av}v∈V be a finite collection of events and G = (V, E) be a chordal graph. Our main result – the chordal graph sieve – is a Bonferroni-type inequality where the selection of intersections in the estimates is determined by a chordal graph G. It interpolates between Boole's inequality (G empty) and the sieve formula (G complete). By varying G, several inequalities both well-known and new are obtained in a concise and unified way.

We show that the limiting distribution of the number of comparisons used by Hoare's quickselect algorithm when given a random permutation of n elements for finding the mth-smallest element, where m = o(n), is the Dickman function. The limiting distribution of the number of exchanges is also derived.

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.

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.

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.

In this note it is shown that resistance and resistance dimension are rough isometry invariant.