If n ⩾ k + 1 and G is a connected n-vertex graph, then one can add $\binom{k}{2}$ edges to G so that the resulting graph contains the complete graph Kk+1. This yields that for any connected graph G with at least k + 1 vertices, one can add $\binom{k}{2}$ edges to G so that the resulting graph has chromatic number > k. A long time ago, Bollobás suggested that for every k ⩾ 3 there exists a k-chromatic graph Gk such that after adding to it any $\binom{k}{2}$ − 1 edges, the chromatic number of the resulting graph is still k. In this note we prove this conjecture.

We give a new combinatorial interpretation of the stationary distribution of the (partially) asymmetric exclusion process on a finite number of sites in terms of decorated alternative trees and coloured permutations. The corresponding expressions of the multivariate partition functions are then related to multivariate generalisations of Eulerian polynomials for coloured permutations considered recently by N. Williams and the third author, and others. We also discuss stability and negative dependence properties satisfied by the partition functions.

For an odd prime $p$, a $p$-transposition group is a group generated by a set of involutions such that the product of any two has order 2 or $p$. We first classify a family of $(G,2)$-geodesic transitive Cayley graphs ${\rm\Gamma}:=\text{Cay}(T,S)$ where $S$ is a set of involutions and $T:\text{Inn}(T)\leq G\leq T:\text{Aut}(T,S)$. In this case, $T$ is either an elementary abelian 2-group or a $p$-transposition group. Then under the further assumption that $G$ acts quasiprimitively on the vertex set of ${\rm\Gamma}$, we prove that: (1) if ${\rm\Gamma}$ is not $(G,2)$-arc transitive, then this quasiprimitive action is the holomorph affine type; (2) if $T$ is a $p$-transposition group and $S$ is a conjugacy class, then $p=3$ and ${\rm\Gamma}$ is $(G,2)$-arc transitive.

This paper contributes to the regular covers of a complete bipartite graph minus a matching, denoted $K_{n,n}-nK_{2}$, whose fiber-preserving automorphism group acts 2-arc-transitively. All such covers, when the covering transformation group $K$ is either cyclic or $\mathbb{Z}_{p}^{2}$ with $p$ a prime, have been determined in Xu and Du [‘2-arc-transitive cyclic covers of $K_{n,n}-nK_{2}$’, J. Algebraic Combin.39 (2014), 883–902] and Xu et al. [‘2-arc-transitive regular covers of $K_{n,n}-nK_{2}$ with the covering transformation group $\mathbb{Z}_{p}^{2}$’, Ars. Math. Contemp.10 (2016), 269–280]. Finally, this paper gives a classification of all such covers for $K\cong \mathbb{Z}_{p}^{3}$ with $p$ a prime.

The counting and (upper) mass dimensions of a set A ⊆ $\mathbb{R}^d$ are

$$D(A) = \limsup_{\|C\| \to \infty} \frac{\log | \lfloor A \rfloor \cap C |}{\log \|C\|}, \quad \smash{\overline{D}}\vphantom{D}(A) = \limsup_{\ell \to \infty} \frac{\log | \lfloor A \rfloor \cap [-\ell,\ell)^d |}{\log (2 \ell)},$$

$\mathbb{R}^d$

$$D(A) = \text{inf} \{ \alpha \geq 0 \mid {d_{H}^{\alpha}}(A) = 0 \},$$

$${d_{H}^{\alpha}}(A) = \lim_{r \rightarrow 0} \limsup_{\|C\| \rightarrow \infty} \inf \biggl\{ \sum_i \biggl(\frac{\|C_i\|}{\|C\|} \biggr)^\alpha \ \bigg| \ 1 \leq \|C_i\| \leq r \|C\| \biggr\}$$

$\mathbb{R}^d$

An old conjecture of Z. Tuza says that for any graph G, the ratio of the minimum size, τ3(G), of a set of edges meeting all triangles to the maximum size, ν3(G), of an edge-disjoint triangle packing is at most 2. Here, disproving a conjecture of R. Yuster, we show that for any fixed, positive α there are arbitrarily large graphs G of positive density satisfying τ3(G) > (1 − o(1))|G|/2 and ν3(G) < (1 + α)|G|/4.

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 Cs3 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(Cs3, Kn3) 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 S1 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 G1 × G2 denote the strong product of graphs G1 and G2, that is, the graph on V(G1) × V(G2) in which (u1, u2) and (v1, v2) are adjacent if for each i = 1, 2 we have ui = vi or uivi ∈ E(Gi). 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 n3/4 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

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

$k=k(q)$

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