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.

Häggkvist and Scott asked whether one can find a quadratic function q(k) such that, if G is a graph of minimum degree at least q(k), then G contains vertex-disjoint cycles of k consecutive even lengths. In this paper, it is shown that if G is a graph of average degree at least k2+19k+10 with sufficiently many vertices, then G contains vertex-disjoint cycles of k consecutive even lengths, answering the above question in the affirmative. The coefficient of k2 cannot be decreased and, in this sense, this result is best possible.

We consider the problem of sampling ‘unlabelled structures’, i.e., sampling combinatorial structures modulo a group of symmetries. The main tool which has been used for this sampling problem is Burnside’s lemma. In situations where a significant proportion of the structures have no nontrivial symmetries, it is already fairly well understood how to apply this tool. More generally, it is possible to obtain nearly uniform samples by simulating a Markov chain that we call the Burnside process: this is a random walk on a bipartite graph which essentially implements Burnside’s lemma. For this approach to be feasible, the Markov chain ought to be ‘rapidly mixing’, i.e., converge rapidly to equilibrium. The Burnside process was known to be rapidly mixing for some special groups, and it has even been implemented in some computational group theory algorithms. In this paper, we show that the Burnside process is not rapidly mixing in general. In particular, we construct an infinite family of permutation groups for which we show that the mixing time is exponential in the degree of the group.

For a stochastic approximation-type recursion with finitely many possible limit points, we find a lower bound on the probability of converging to a prescribed point in its ‘domain of attraction’. This has implications for the lock-in phenomena in the stochastic models of increasing return economics and the sample complexity of stochastic approximation algorithms in engineering.

Suppose that G is a graph with maximum degree d(G) such that, for every vertex v in G, the neighbourhood of v contains at most d(G)2/f (f > 1) edges. We show that the list chromatic number of G is at most Kd(G)/log f, for some positive constant K. This result is sharp up to the multiplicative constant K and strengthens previous results by Kim [9], Johansson [7], Alon, Krivelevich and Sudakov [3], and the present author [18]. This also motivates several interesting questions.

As an application, we derive several upper bounds for the strong (list) chromatic index of a graph, under various assumptions. These bounds extend earlier results by Faudree, Gyárfás, Schelp and Tuza [6] and Mahdian [13] and determine, up to a constant factor, the strong (list) chromatic index of a random graph. Another application is an extension of a result of Kostochka and Steibitz [10] concerning the structure of list critical graphs.

Given an r-graph F, an r-graph G is called weakly F-saturated if the edges missing from G can be added, one at a time, in some order, each extra edge creating a new copy of F. Let w-sat(n, F) be the minimal size of a weakly F-saturated graph of order n. We compute the w-sat function for a wide class of r-graphs called pyramids: these include many examples for which the w-sat function was known, as well as many new examples, such as the computation of w-sat(n,Ks + Kt), and enable us to prove a conjecture of Tuza.

Our main technique, developed from ideas of Kalai, is based on matroids derived from exterior algebra. We prove that if it succeeds for some graphs then the same is true for the ‘cones’ and ‘joins’ of such graphs, so that the w-sat function can be computed for many graphs that are built up from certain elementary graphs by these operations.

Let G be a graph on vertex set [n], and for X ⊆ [n] let N(X) be the union of X and its neighbourhood in G. A family of sets [Fscr] ⊆ 2[n] is G-intersecting if N(X) ∩ Y ≠ [empty ] for all X, Y ∈ [Fscr]. In this paper we study the cardinality and structure of the largest k-uniform G-intersecting families on a fixed graph G.

In this paper we prove the following almost optimal theorem. For any δ > 0, there exist constants c and n0 such that, if n [ges ] n0, T is a tree of order n and maximum degree at most cn/log n, and G is a graph of order n and minimum degree at least (1/2 + δ)n, then T is a subgraph of G.

We generalize a minimal 3-connectivity result of Halin from graphs to binary matroids. As applications of this theorem to minimally 3-connected matroids, we obtain new results and short inductive proofs of results of Oxley and Wu. We also give new short inductive proofs of results of Dirac and Halin on minimally k-connected graphs for k ∈ {2,3}.

We strengthen the Cohen–Körner–Simonyi upper bound on counting ‘very different sequences’.

We introduce a notion of the derivative with respect to a function, not necessarily related to a probability distribution, which generalizes the concept of derivative as proposed by Lebesgue [14]. The differential calculus required to solve linear differential equations using this notion of the derivative is included in the paper. The definition given here may also be considered as the inverse operator of a modified notion of the Riemann–Stieltjes integral. Both this unified approach and the results of differential calculus allow us to characterize distributions in terms of three different types of conditional expectations. In applying these results, a test of goodness of fit is also indicated. Finally, two characterizations of a general Poisson process are included. Specifically, a useful result for the homogeneous Poisson process is generalized.

A random interval graph of order n is generated by picking 2n numbers X1,…,X2n independently from the uniform distribution on [0,1] and considering the collection of n intervals with endpoints X2i−1 and X2i for i ∈ {1,…,n}. The graph vertices correspond to intervals. Two vertices are connected if the corresponding intervals intersect. This paper characterizes the fluctuations of the independence number in random interval graphs. This characterization is obtained through the analysis of the greedy algorithm. We actually prove limit theorems (central limit theorem and large deviation principle) on the number of phases of this greedy algorithm. The proof relies on the analysis of first-passage times through a random level.

Let [Mscr] be the class of simple matroids which do not contain the 5-point line U2,5, the Fano plane F7, the non-Fano plane F−7, or the matroid P7 as minors. Let h(n) be the maximum number of points in a rank-n matroid in [Mscr]. We show that h(2) = 4, h(3) = 7, and h(n) = (n+12) for n [ges ] 4, and we also find all the maximum-sized matroids for each rank.

Algorithmic aspects of a chip-firing game on a graph introduced by Biggs are studied. This variant of the chip-firing game, called the dollar game, has the properties that every starting configuration leads to a so-called critical configuration. The set of critical configurations has many interesting properties. In this paper it is proved that the number of steps needed to reach a critical configuration is polynomial in the number of edges of the graph and the number of chips in the starting configuration, but not necessarily in the size of the input. An alternative algorithm is also described and analysed.

We consider an extension of the Monotone Subsequence lemma of Erdős and Szekeres in higher dimensions. Let v1,…,vn ∈ ℝd be a sequence of real vectors. For a subset I ⊆ [n] and vector [srarr ]c ∈ {0,1}d we say that I is [srarr ]c-free if there are no i < j ∈ I, such that, for every k = 1,…,d, vik < vik if and only if [srarr ]ck = 0. We construct sequences of vectors with the property that the largest [srarr ]c-free subset is small for every choice of [srarr ]c. In particular, for d = 2 the largest [srarr ]c-free subset is O(n⅝) for all the four possible [srarr ]c. The smallest possible value remains far from being determined.

We also consider and resolve a simpler variant of the problem.

We investigate a graph function which is related to the local density, the maximal cut and the least eigenvalue of a graph. In particular it enables us to prove the following assertions.

Let p [ges ] 3 be an integer, c ∈ (0, 1/2) and G be a Kp-free graph on n vertices with e [les ] cn2 edges. There exists a positive constant α = α (c, p) such that:

(a) some [lfloor]n/2[rfloor]-subset of V (G) induces at most (c-4 − α) n2 edges (this answers a question of Paul Erdős);

(b) G can be made bipartite by the omission of at most (c-2 − α) n2 edges.

We shall pack circuits of arbitrary lengths into the complete graph KN. More precisely, if N is odd and [sum ]ti=1mi = (N2), mi [ges ] 3, then the edges of KN can be written as an edge-disjoint union of circuits of lengths m1,…,mt. Since the degrees of the vertices in any such packing must be even, this result cannot hold for even N. For N even, we prove that if [sum ]ti=1mi [les ] (N2) − N−2 then we can write some subgraph of KN as an edge-disjoint union of circuits of lengths m1,…,mt. In particular, KN minus a 1-factor can be written as a union of such circuits when [sum ]ti=1mi = (N2) − N−2. We shall also show that these results are best possible.