The results of this paper concern the ‘large spectra’ of sets, by which we mean the set of points in ${\bb F}_p^{\times}$ at which the Fourier transform of a characteristic function $\chi_A$, $A\subseteq {\bb F}_p$, can be large. We show that a recent result of Chang concerning the structure of the large spectrum is best possible. Chang's result has already found a number of applications in combinatorial number theory.

We also show that if $|A|=\lfloor {p/2}\rfloor$, and if $R$ is the set of points $r$ for which $|\hat{\chi}_A(r)|\geqslant \alpha p$, then almost nothing can be said about $R$ other than that $|R|\ll \alpha^{-2}$, a trivial consequence of Parseval's theorem.

Let G be a fixed connected multigraph with no loops. A random n-lift of G is obtained by replacing each vertex of G by a set of n vertices (where these sets are pairwise disjoint) and replacing each edge by a randomly chosen perfect matching between the n-sets corresponding to the endpoints of the edge. Let XG be the number of perfect matchings in a random lift of G. We study the distribution of XG in the limit as n tends to infinity, using the small subgraph conditioning method.

We present several results including an asymptotic formula for the expectation of XG when G is d-regular, d ≥ 3. The interaction of perfect matchings with short cycles in random lifts of regular multigraphs is also analysed. Partial calculations are performed for the second moment of XG, with full details given for two example multigraphs, including the complete graph K4.

To assist in our calculations we provide a theorem for estimating a summation over multiple dimensions using Laplace's method. This result is phrased as a summation over lattice points, and may prove useful in future applications.

Let q be an integer with q [ges ] 2. We give a new proof of a result of Erdös and Turán determining the proportion of elements of the finite symmetric group Sn having no cycle of length a multiple of q. We then extend our methods to the more difficult case of obtaining the proportion of such elements in the finite alternating group An. In both cases, we derive an asymptotic formula with error term for the above mentioned proportion, which contains an unexpected occurrence of the Gamma-function.

We apply these results to estimate the proportion of elements of order 2f in Sn, and of order 3f in An and Sn, where gcd(2, f) = 1, and gcd(3, f) = 1, respectively, and log f is polylogarithmic in n. We also give estimates for the probability that the fth power of such elements is a transposition or a 3-cycle, respectively. An algorithmic application of these results to computing in An or Sn, given as a black-box group with an order oracle, is discussed.

We investigate the rank of random (symmetric) sparse matrices. Our main finding is that with high probability, any dependency that occurs in such a matrix is formed by a set of few rows that contains an overwhelming number of zeros. This allows us to obtain an exact estimate for the co-rank.

Gerards and Seymour (see [10], p. 115) conjectured that if a graph has no odd complete minor of order l, then it is (l − 1)-colourable. This is an analogue of the well-known conjecture of Hadwiger, and in fact, this would immediately imply Hadwiger's conjecture. The current best-known bound for the chromatic number of graphs with no odd complete minor of order l is by the recent result by Geelen, Gerards, Reed, Seymour and Vetta [8], and by Kawarabayashi [12] later, independently. But it seems very hard to improve this bound since this would also improve the current best-known bound for the chromatic number of graphs with no complete minor of order l.

Motivated by this problem, in this note we show that there exists an absolute constant f(k) such that any graph G with no odd complete minor of order k admits a vertex partition V1, . . ., V496k such that each component in the subgraph induced on Vi (i ≥ 1) has at most f(k) vertices. When f(k) = 1, this is a colouring of G. Hence this is a relaxation of colouring in a sense, and this is the first result in this direction for the odd Hadwiger's conjecture.

Our proof is based on a recent decomposition theorem due to Geelen, Gerards, Reed, Seymour and Vetta [8], together with a connectivity result that forces a huge complete bipartite minor in large graphs by Böhme, Kawarabayashi, Maharry and Mohar [3].

Let Δ ≥ 3 be an integer. Given a fixed z ∈ +Δ such that zΔ > 0, we consider a graph Gz drawn uniformly at random from the collection of graphs with zin vertices of degree i for i = 1,. . .,Δ. We study the performance of the Karp–Sipser algorithm when applied to Gz. If there is an index δ > 1 such that z1 = . . . = zδ−1 = 0 and δzδ,. . .,ΔzΔ is a log-concave sequence of positive reals, then with high probability the Karp–Sipser algorithm succeeds in finding a matching with n ∥ z ∥ 1/2 − o(n1−ε) edges in Gz, where ε = ε (Δ, z) is a constant.

We show that any n-vertex complete graph with edges coloured with three colours contains a set of at most four vertices such that the number of the neighbours of these vertices in one of the colours is at least 2n/3. The previous best value, proved by Erdős, Faudree, Gould, Gyárfás, Rousseau and Schelp in 1989, is 22. It is conjectured that three vertices suffice.

Jackson [10] gave a polynomial sufficient condition for a bipartite tournament to contain a cycle of a given length. The question arises as to whether deciding on the maximum length of a cycle in a bipartite tournament is polynomial. The problem was considered by Manoussakis [12] in the slightly more general setting of 2-edge coloured complete graphs: is it polynomial to find a longest alternating cycle in such coloured graphs? In this paper, strong evidence is given that such an algorithm exists. In fact, using a reduction to the well known exact matching problem, we prove that the problem is random polynomial.

We show by elementary methods that given any finite partition of the set ℕ of positive integers, there is one cell that is both additively and multiplicatively rich. In particular, this cell must contain a sequence and all of its finite sums, and another sequence and all of its finite products, a fact that was previously known only by utilizing the algebraic structure of the Stone–Čech compactification βℕ of ℕ.

We follow the example of Tutte in his construction of the dichromate of a graph (i.e. the Tutte polynomial) as a unification of the chromatic polynomial and the flow polynomial in order to construct a new polynomial invariant of maps (graphs embedded in orientable surfaces). We call this the surface Tutte polynomial. The surface Tutte polynomial of a map contains the Las Vergnas polynomial, the Bollobás–Riordan polynomial and the Krushkal polynomial as specializations. By construction, the surface Tutte polynomial includes among its evaluations the number of local tensions and local flows taking values in any given finite group. Other evaluations include the number of quasi-forests.

We derive the distribution of the number of links and the average weight for the shortest path tree (SPT) rooted at an arbitrary node to $m$ uniformly chosen nodes in the complete graph of size $N$ with i.i.d. exponential link weights. We rely on the fact that the full shortest path tree to all destinations (ie, $m=N-1$) is a uniform recursive tree to derive a recursion for the generating function of the number of links of the SPT, and solve this recursion exactly.

The explicit form of the generating function allows us to compute the expectation and variance of the size of the subtree for all $m$. We also obtain exact expressions for the average weight of the subtree.

Let 0 < p < 1, q = 1 − p and b be fixed and let G ∈ [Gscr ](n, p) be a graph on n vertices where each pair of vertices is joined independently with probability p. We show that the probability that every vertex of G has degree at most pn + b √npq is equal to (c + o(1))n, for c = c(b) the root of a certain equation. Surprisingly, c(0) = 0.6102 … is greater than ½ and c(b) is independent of p. To obtain these results we consider the complete graph on n vertices with weights on the edges. Taking these weights as independent normal N(p, pq) random variables gives a ‘continuous’ approximation to [Gscr ](n, p) whose degrees are much easier to analyse.

We consider the popular and well-studied push model, which is used to spread information in a given network with n vertices. Initially, some vertex owns a rumour and passes it to one of its neighbours, which is chosen randomly. In each of the succeeding rounds, every vertex that knows the rumour informs a random neighbour. It has been shown on various network topologies that this algorithm succeeds in spreading the rumour within O(log n) rounds. However, many studies are quite coarse and involve huge constants that do not allow for a direct comparison between different network topologies. In this paper, we analyse the push model on several important families of graphs, and obtain tight runtime estimates. We first show that, for any almost-regular graph on n vertices with small spectral expansion, rumour spreading completes after log2n + log n+o(log n) rounds with high probability. This is the first result that exhibits a general graph class for which rumour spreading is essentially as fast as on complete graphs. Moreover, for the random graph G(n,p) with p=c log n/n, where c > 1, we determine the runtime of rumour spreading to be log2n + γ (c)log n with high probability, where γ(c) = clog(c/(c−1)). In particular, this shows that the assumption of almost regularity in our first result is necessary. Finally, for a hypercube on n=2d vertices, the runtime is with high probability at least (1+β) ⋅ (log2n + log n), where β > 0. This reveals that the push model on hypercubes is slower than on complete graphs, and thus shows that the assumption of small spectral expansion in our first result is also necessary. In addition, our results combined with the upper bound of O(log n) for the hypercube (see [11]) imply that the push model is faster on hypercubes than on a random graph G(n, clog n/n), where c is sufficiently close to 1.

Let the Kp-independence number αp (G) of a graph G be the maximum order of an induced subgraph in G that contains no Kp. (So K2-independence number is just the maximum size of an independent set.) For given integers r, p, m > 0 and graphs L1,…,Lr, we define the corresponding Turán-Ramsey function RTp(n, L1,…,Lr, m) to be the maximum number of edges in a graph Gn of order n such that αp(Gn) ≤ m and there is an edge-colouring of G with r colours such that the jth colour class contains no copy of Lj, for j = 1,…, r. In this continuation of [11] and [12], we will investigate the problem where, instead of α(Gn) = o(n), we assume (for some fixed p > 2) the stronger condition that αp(Gn) = o(n). The first part of the paper contains multicoloured Turán-Ramsey theorems for graphs Gn of order n with small Kp-independence number αp(Gn). Some structure theorems are given for the case αp(Gn) = o(n), showing that there are graphs with fairly simple structure that are within o(n2) of the extremal size; the structure is described in terms of the edge densities between certain sets of vertices.

The second part of the paper is devoted to the case r = 1, i.e., to the problem of determining the asymptotic value of

for p < q. Several results are proved, and some other problems and conjectures are stated.

We show how to generate labelled and unlabelled outerplanar graphs with $n$ vertices uniformly at random in polynomial time in $n$. To generate labelled outerplanar graphs, we present a counting technique using the decomposition of a graph according to its block structure, and compute the exact number of labelled outerplanar graphs. This allows us to make the correct probabilistic choices in a recursive generation of uniformly distributed outerplanar graphs.

Next we modify our formulas to also count rooted unlabelled graphs, and finally show how to use these formulas in a Las Vegas algorithm to generate unlabelled outerplanar graphs uniformly at random in expected polynomial time.

We show that a set A ⊂ {0, 1}n with edge-boundary of size at most can be made into a subcube by at most (2ε/log2(1/ε))|A| additions and deletions, provided ε is less than an absolute constant.

We deduce that if A ⊂ {0, 1}n has size 2t for some t ∈ ℕ, and cannot be made into a subcube by fewer than δ|A| additions and deletions, then its edge-boundary has size at least provided δ is less than an absolute constant. This is sharp whenever δ = 1/2j for some j ∈ {1, 2, . . ., t}.

We study the fraction of time that a Markov chain spends in a given subset of states. We give an exponential bound on the probability that it exceeds its expectation by a constant factor. Our bound depends on the mixing properties of the chain, and is asymptotically optimal for a certain class of Markov chains. It beats the best previously known results in this direction. We present an application to the leader election problem.

We say that a (di)graph G has a perfect H-packing if there exists a set of vertex-disjoint copies of H which cover all the vertices in G. The seminal Hajnal–Szemerédi theorem characterizes the minimum degree that ensures a graph G contains a perfect Kr-packing. In this paper we prove the following analogue for directed graphs: Suppose that T is a tournament on r vertices and G is a digraph of sufficiently large order n where r divides n. If G has minimum in- and outdegree at least (1−1/r)n then G contains a perfect T-packing.

In the case when T is a cyclic triangle, this result verifies a recent conjecture of Czygrinow, Kierstead and Molla [4] (for large digraphs). Furthermore, in the case when T is transitive we conjecture that it suffices for every vertex in G to have sufficiently large indegree or outdegree. We prove this conjecture for transitive triangles and asymptotically for all r ⩾ 3. Our approach makes use of a result of Keevash and Mycroft [10] concerning almost perfect matchings in hypergraphs as well as the Directed Graph Removal Lemma [1, 6].

We show that a random 4-regular graph asymptotically almost surely (a.a.s.) has chromatic number 3. The proof uses an efficient algorithm which a.a.s. 3-colours a random 4-regular graph. The analysis includes use of the differential equation method, and exponential bounds on the tail of random variables associated with branching processes. A substantial part of the analysis applies to random d-regular graphs in general.

In this paper we analyse the expected depth of random circuits of fixed fanin f. Such circuits are built a gate at a time, with the f inputs of each new gate being chosen randomly from among the previously added gates. The depth of the new gate is defined to be one more than the maximal depth of its input gates. We show that the expected depth of a random circuit with n gates is bounded from above by ef ln n and from below by 2.04 … f ln n.