We give several results showing that different discrete structures typically gain certain spanning substructures (in particular, Hamilton cycles) after a modest random perturbation. First, we prove that adding linearly many random edges to a dense k-uniform hypergraph ensures the (asymptotically almost sure) existence of a perfect matching or a loose Hamilton cycle. The proof involves an interesting application of Szemerédi's Regularity Lemma, which might be independently useful. We next prove that digraphs with certain strong expansion properties are pancyclic, and use this to show that adding a linear number of random edges typically makes a dense digraph pancyclic. Finally, we prove that perturbing a certain (minimum-degree-dependent) number of random edges in a tournament typically ensures the existence of multiple edge-disjoint Hamilton cycles. All our results are tight.

The 3-uniform tight cycle Cs 3 has vertex set ${\mathbb Z}_s$ and edge set {{i, i + 1, i + 2}: i ∈ ${\mathbb Z}_s$ }. We prove that for every s ≢ 0 (mod 3) with s ⩾ 16 or s ∈ {8, 11, 14} there is a cs > 0 such that the 3-uniform hypergraph Ramsey number r(Cs 3, Kn 3) satisfies

$$\begin{equation*}r(C_s^3, K_n^3)< 2^{c_s n \log n}.\\end{equation*}$$

$2^{n^{1+\epsilon_s}}$

We apply the concept of braiding sequences to link polynomials to show polynomial growth bounds on the derivatives of the Jones polynomial evaluated on S 1 and of the Brandt–Lickorish–Millett–Ho polynomial evaluated on [–2, 2] on alternating and positive knots of given genus. For positive links, boundedness criteria for the coefficients of the Jones, HOMFLY and Kauffman polynomials are derived. (This is a continuation of the paper ‘Applications of braiding sequences. I’: Commun. Contemp. Math.12(5) (2010), 681–726.)

Let G 1 × G 2 denote the strong product of graphs G 1 and G 2, that is, the graph on V(G 1) × V(G 2) in which (u 1, u 2) and (v 1, v 2) are adjacent if for each i = 1, 2 we have ui = vi or u i v i ∈ E(G i ). The Shannon capacity of G is c(G) = limn → ∞ α(Gn )1/n , where Gn denotes the n-fold strong power of G, and α(H) denotes the independence number of a graph H. The normalized Shannon capacity of G is

$$C(G) = \ffrac {\log c(G)}{\log |V(G)|}.$$

We study cup products in the integral cohomology of the Hilbert scheme of $n$ points on a K3 surface and present a computer program for this purpose. In particular, we deal with the question of which classes can be represented by products of lower degrees.

Supplementary materials are available with this article.

Working in the infinite plane R2, consider a Poisson process of black points with intensity 1, and an independent Poisson process of red points with intensity λ. We grow a disc around each black point until it hits the nearest red point, resulting in a random configuration Aλ, which is the union of discs centered at the black points. Next, consider a fixed disc of area n in the plane. What is the probability pλ(n) that this disc is covered by Aλ? We prove that if λ3nlogn = y then, for sufficiently large n, e-8π2y ≤ pλ(n) ≤ e-2π2y/3. The proofs reveal a new and surprising phenomenon, namely, that the obstructions to coverage occur on a wide range of scales.

The point-line collinearity graph ${\mathcal{G}}$ of the maximal 2-local geometry for the largest simple Fischer group, $Fi_{24}^{\prime }$ , is extensively analysed. For an arbitrary vertex $a$ of ${\mathcal{G}}$ , the $i\text{th}$ -disc of $a$ is described in detail. As a consequence, it follows that ${\mathcal{G}}$ has diameter $5$ . The collapsed adjacency matrix of ${\mathcal{G}}$ is given as well as accompanying computer files which contain a wealth of data about ${\mathcal{G}}$ .

Supplementary materials are available with this article.

A graph is called arc-regular if its full automorphism group acts regularly on its arc set. In this paper, we completely determine all the arc-regular Frobenius metacirculants of prime valency.

We introduce a new model of competition on growing networks. This extends the preferential attachment model, with the key property that node choices evolve simultaneously with the network. When a new node joins the network, it chooses neighbours by preferential attachment, and selects its type based on the number of initial neighbours of each type. The model is analysed in detail, and in particular, we determine the possible proportions of the various types in the limit of large networks. An important qualitative feature we find is that, in contrast to many current theoretical models, often several competitors will coexist. This matches empirical observations in many real-world networks.

Recent inapproximability results of Sly (2010), together with an approximation algorithm presented by Weitz (2006), establish a beautiful picture of the computational complexity of approximating the partition function of the hard-core model. Let λc ( $\mathbb{T}_{\Delta}$ ) denote the critical activity for the hard-model on the infinite Δ-regular tree. Weitz presented an FPTAS for the partition function when λ < λc ( $\mathbb{T}_{\Delta}$ ) for graphs with constant maximum degree Δ. In contrast, Sly showed that for all Δ ⩾ 3, there exists εΔ > 0 such that (unless RP = NP) there is no FPRAS for approximating the partition function on graphs of maximum degree Δ for activities λ satisfying λc ( $\mathbb{T}_{\Delta}$ ) < λ < λc ( $\mathbb{T}_{\Delta}$ ) + εΔ.

We prove that a similar phenomenon holds for the antiferromagnetic Ising model. Sinclair, Srivastava and Thurley (2014) extended Weitz's approach to the antiferromagnetic Ising model, yielding an FPTAS for the partition function for all graphs of constant maximum degree Δ when the parameters of the model lie in the uniqueness region of the infinite Δ-regular tree. We prove the complementary result for the antiferromagnetic Ising model without external field, namely, that unless RP = NP, for all Δ ⩾ 3, there is no FPRAS for approximating the partition function on graphs of maximum degree Δ when the inverse temperature lies in the non-uniqueness region of the infinite tree $\mathbb{T}_{\Delta}$ . Our proof works by relating certain second moment calculations for random Δ-regular bipartite graphs to the tree recursions used to establish the critical points on the infinite tree.

We extend spectral graph theory from the integral circulant graphs with prime power order to a Cayley graph over a finite chain ring and determine the spectrum and energy of such graphs. Moreover, we apply the results to obtain the energy of some gcd-graphs on a quotient ring of a unique factorisation domain.

We propose a new method, probabilistic divide-and-conquer, for improving the success probability in rejection sampling. For the example of integer partitions, there is an ideal recursive scheme which improves the rejection cost from asymptotically order n 3/4 to a constant. We show other examples for which a non-recursive, one-time application of probabilistic divide-and-conquer removes a substantial fraction of the rejection sampling cost.

We also present a variation of probabilistic divide-and-conquer for generating i.i.d. samples that exploits features of the coupon collector's problem, in order to obtain a cost that is sublinear in the number of samples.

We provide a comprehensive study of the function $h=h(q)$ defined by

$$\begin{eqnarray}h=q\mathop{\prod }_{j=1}^{\infty }\frac{(1-q^{12j-1})(1-q^{12j-11})}{(1-q^{12j-5})(1-q^{12j-7})}\end{eqnarray}$$

$k=k(q)$

$$\begin{eqnarray}k=q\mathop{\prod }_{j=1}^{\infty }\frac{(1-q^{10j-1})(1-q^{10j-2})(1-q^{10j-8})(1-q^{10j-9})}{(1-q^{10j-3})(1-q^{10j-4})(1-q^{10j-6})(1-q^{10j-7})}.\end{eqnarray}$$

A uniform hypergraph H is called k-Ramsey for a hypergraph F if, no matter how one colours the edges of H with k colours, there is always a monochromatic copy of F. We say that H is k-Ramsey-minimal for F if H is k-Ramsey for F but every proper subhypergraph of H is not. Burr, Erdős and Lovasz studied various parameters of Ramsey-minimal graphs. In this paper we initiate the study of minimum degrees and codegrees of Ramsey-minimal 3-uniform hypergraphs. We show that the smallest minimum vertex degree over all k-Ramsey-minimal 3-uniform hypergraphs for Kt (3) is exponential in some polynomial in k and t. We also study the smallest possible minimum codegree over 2-Ramsey-minimal 3-uniform hypergraphs.

In a recent paper, Baxter and Zeilberger showed that the two most important Mahonian statistics, the inversion number and the major index, are asymptotically independently normally distributed on permutations. In another recent paper, Canfield, Janson and Zeilberger proved the result, already known to statisticians, that the Mahonian distribution is asymptotically normal on words. This leaves one question unanswered: What, asymptotically, is the joint distribution of the inversion number and the major index on words? We answer this question by establishing convergence to a bivariate normal distribution.

Zhou and Feng [‘On symmetric graphs of valency five’, Discrete Math. 310 (2010), 1725–1732] proved that all connected pentavalent 1-transitive Cayley graphs of finite nonabelian simple groups are normal. We construct an example of a nonnormal 2-arc transitive pentavalent symmetric Cayley graph on the alternating group $\text{A}_{39}$. Furthermore, we show that the full automorphism group of this graph is isomorphic to the alternating group $\text{A}_{40}$.

We continue our study of intermediate sums over polyhedra, interpolating between integrals and discrete sums, which were introduced by Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex. Math. Comp. 75 (2006), 1449–1466]. By well-known decompositions, it is sufficient to consider the case of affine cones $s+\mathfrak{c}$ , where $s$ is an arbitrary real vertex and $\mathfrak{c}$ is a rational polyhedral cone. For a given rational subspace  $L$ , we define the intermediate generating functions $S^{L}(s+\mathfrak{c})(\unicode[STIX]{x1D709})$ by integrating an exponential function over all lattice slices of the affine cone  $s+\mathfrak{c}$ parallel to the subspace  $L$ and summing up the integrals. We expose the bidegree structure in parameters $s$ and $\unicode[STIX]{x1D709}$ , which was implicitly used in the algorithms in our papers [Computation of the highest coefficients of weighted Ehrhart quasi-polynomials of rational polyhedra. Found. Comput. Math.12 (2012), 435–469] and [Intermediate sums on polyhedra: computation and real Ehrhart theory. Mathematika59 (2013), 1–22]. The bidegree structure is key to a new proof for the Baldoni–Berline–Vergne approximation theorem for discrete generating functions [Local Euler–Maclaurin expansion of Barvinok valuations and Ehrhart coefficients of rational polytopes. Contemp. Math.452 (2008), 15–33], using the Fourier analysis with respect to the parameter  $s$ and a continuity argument. Our study also enables a forthcoming paper, in which we study intermediate sums over multi-parameter families of polytopes.

An orthogonal coloring of the two-dimensional unit sphere $\mathbb{S}^{2}$ , is a partition of $\mathbb{S}^{2}$ into parts such that no part contains a pair of orthogonal points: that is, a pair of points at spherical distance ${\it\pi}/2$ apart. It is a well-known result that an orthogonal coloring of $\mathbb{S}^{2}$ requires at least four parts, and orthogonal colorings with exactly four parts can easily be constructed from a regular octahedron centered at the origin. An intriguing question is whether or not every orthogonal 4-coloring of $\mathbb{S}^{2}$ is such an octahedral coloring. In this paper, we address this question and show that if every color class has a non-empty interior, then the coloring is octahedral. Some related results are also given.

We relate Catalan numbers and Catalan determinants to random matrix integrals and to moments of spin representations of odd orthogonal groups.