In the full rectangular version of Gilbert's planar tessellation (see Gilbert (1967), Mackisack and Miles (1996), and Burridge et al. (2013)), lines extend either horizontally (with east- and west-growing rays) or vertically (north- and south-growing rays) from seed points which form a stationary Poisson point process, each ray stopping when it meets another ray that has blocked its path. In the half-Gilbert rectangular version, east- and south-growing rays, whilst having the potential to block each other, do not interact with west and north rays, and vice versa. East- and south-growing rays have a reciprocality of blocking, as do west and north. In this paper we significantly expand and simplify the half-Gilbert analytic results that we gave in Burridge et al. (2013). We also show how the idea of reciprocality of blocking between rays can be used in a much wider context, with rays not necessarily orthogonal and with seeds that produce more than two rays.

In their recent paper Velleman and Warrington (2013) analyzed the expected values of some of the parameters in a memory game; namely, the length of the game, the waiting time for the first match, and the number of lucky moves. In this paper we continue this direction of investigation and obtain the limiting distributions of those parameters. More specifically, we prove that when suitably normalized, these quantities converge in distribution to a normal, Rayleigh, and Poisson random variable, respectively. We also make a connection between the memory game and one of the models of preferential attachment graphs. In particular, as a by-product of our methods, we obtain the joint asymptotic normality of the degree counts in the preferential attachment graphs. Furthermore, we obtain simpler proofs (although without rate of convergence) of some of the results of Peköz et al. (2014) on the joint limiting distributions of the degrees of the first few vertices in preferential attachment graphs. In order to prove that the length of the game is asymptotically normal, our main technical tool is a limit result for the joint distribution of the number of balls in a multitype generalized Pólya urn model.

We consider a directed graph on the integers with a directed edge from vertex i to j present with probability n-1, whenever i < j, independently of all other edges. Moreover, to each edge (i, j) we assign weight n-1(j - i). We show that the closure of vertex 0 in such a weighted random graph converges in distribution to the Poisson-weighted infinite tree as n → ∞. In addition, we derive limit theorems for the length of the longest path in the subgraph of the Poisson-weighted infinite tree which has all vertices at weighted distance of at most ρ from the root.

We show that certain topologically defined uniform spanning tree probabilities for graphs embedded in an annulus can be computed as linear combinations of Pfaffians of matrices involving the line-bundle Green's function, where the coefficients count cover-inclusive Dyck tilings of skew Young diagrams.

We investigate properties which ensure that a given finite graph is the commuting graph of a group or semigroup. We show that all graphs on at least two vertices such that no vertex is adjacent to all other vertices is the commuting graph of some semigroup. Moreover, we obtain complete classifications of the graphs with an isolated vertex or edge that are the commuting graph of a group and the cycles that are the commuting graph of a centrefree semigroup.

We consider the problem of minimizing the number of triangles in a graph of given order and size, and describe the asymptotic structure of extremal graphs. This is achieved by characterizing the set of flag algebra homomorphisms that minimize the triangle density.

A long-standing conjecture of Richter and Thomassen states that the total number of intersection points between any n simple closed Jordan curves in the plane, so that any pair of them intersect and no three curves pass through the same point, is at least (1−o(1))n 2.

We confirm the above conjecture in several important cases, including the case (1) when all curves are convex, and (2) when the family of curves can be partitioned into two equal classes such that each curve from the first class touches every curve from the second class. (Two closed or open curves are said to be touching if they have precisely one point in common and at this point the two curves do not properly cross.)

An important ingredient of our proofs is the following statement. Let S be a family of n open curves in ℝ2, so that each curve is the graph of a continuous real function defined on ℝ, and no three of them pass through the same point. If there are nt pairs of touching curves in S, then the number of crossing points is $\Omega(nt\sqrt{\log t/\log\log t})$ .

Random increasing k-trees represent an interesting and useful class of strongly dependent graphs that have been studied widely, including being used recently as models for complex networks. In this paper we study an informative notion called BFS-profile and derive, by several analytic means, asymptotic estimates for its expected value, together with the limiting distribution in certain cases; some interesting consequences predicting more precisely the shapes of random k-trees are also given. Our methods of proof rely essentially on a bijection between k-trees and ordinary trees, the resolution of linear systems, and a specially framed notion called Flajolet–Odlyzko admissibility.

We present a sufficient condition for a pair of finite integer sequences to be degree sequences of a bipartite graph, based only on the lengths of the sequences and their largest and smallest elements.

We show that, for each fixed k, an n-vertex graph not containing a cycle of length 2k has at most $80\sqrt{k\log k}\cdot n^{1+1/k}+O(n)$ edges.

A simple graph $G=(V,E)$ admits an $H$ -covering if every edge in $E$ belongs to at least one subgraph of $G$ isomorphic to a given graph $H$ . Then the graph $G$ is $(a,d)$ - $H$ -antimagic if there exists a bijection $f:V\cup E\rightarrow \{1,2,\ldots ,|V|+|E|\}$ such that, for all subgraphs $H^{\prime }$ of $G$ isomorphic to $H$ , the $H^{\prime }$ -weights, $wt_{f}(H^{\prime })=\sum _{v\in V(H^{\prime })}f(v)+\sum _{e\in E(H^{\prime })}f(e)$ , form an arithmetic progression with the initial term $a$ and the common difference $d$ . When $f(V)=\{1,2,\ldots ,|V|\}$ , then $G$ is said to be super $(a,d)$ - $H$ -antimagic. In this paper, we study super $(a,d)$ - $H$ -antimagic labellings of a disjoint union of graphs for $d=|E(H)|-|V(H)|$ .

In this paper, we count a dual set of Stirling permutations by the number of alternating runs and study properties of the generating functions, including recurrence relations, grammatical interpretations and convolution formulas.

Building on coprincipal mesoprimary decomposition [Kahle and Miller, Decompositions of commutative monoid congruences and binomial ideals, Algebra and Number Theory 8 (2014), 1297–1364], we combinatorially construct an irreducible decomposition of any given binomial ideal. In a parallel manner, for congruences in commutative monoids we construct decompositions that are direct combinatorial analogues of binomial irreducible decompositions, and for binomial ideals we construct decompositions into ideals that are as irreducible as possible while remaining binomial. We provide an example of a binomial ideal that is not an intersection of irreducible binomial ideals, thus answering a question of Eisenbud and Sturmfels [Binomial ideals, Duke Math. J. 84 (1996), 1–45].

Let Qd denote the hypercube of dimension d. Given d ⩾ m, a spanning subgraph G of Qd is said to be (Qd , Qm )-saturated if it does not contain Qm as a subgraph but adding any edge of E(Qd )\E(G) creates a copy of Qm in G. Answering a question of Johnson and Pinto [27], we show that for every fixed m ⩾ 2 the minimum number of edges in a (Qd , Qm )-saturated graph is Θ(2d ).

We also study weak saturation, which is a form of bootstrap percolation. A spanning subgraph of Qd is said to be weakly (Qd , Qm )-saturated if the edges of E(Qd )\E(G) can be added to G one at a time so that each added edge creates a new copy of Qm . Answering another question of Johnson and Pinto [27], we determine the minimum number of edges in a weakly (Qd , Qm )-saturated graph for all d ⩾ m ⩾ 1. More generally, we determine the minimum number of edges in a subgraph of the d-dimensional grid Pkd which is weakly saturated with respect to ‘axis aligned’ copies of a smaller grid Prm . We also study weak saturation of cycles in the grid.

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 K k+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.