Let Dk denote the tournament on 3k vertices consisting of three disjoint vertex classes V1, V2 and V3 of size k, each oriented as a transitive subtournament, and with edges directed from V1 to V2, from V2 to V3 and from V3 to V1. Fox and Sudakov proved that given a natural number k and ε > 0, there is n0(k, ε) such that every tournament of order n ⩾ n0(k,ε) which is ε-far from being transitive contains Dk as a subtournament. Their proof showed that $n_0(k,\epsilon ) \leq \epsilon ^{-O(k/\epsilon ^2)}$ and they conjectured that this could be reduced to n0(k, ε) ⩽ ε−O(k). Here we prove this conjecture.

It is proved that the median eigenvalues of every connected bipartite graph G of maximum degree at most three belong to the interval [−1, 1] with a single exception of the Heawood graph, whose median eigenvalues are $\pm\sqrt{2}$. Moreover, if G is not isomorphic to the Heawood graph, then a positive fraction of its median eigenvalues lie in the interval [−1, 1]. This surprising result has been motivated by the problem about HOMO-LUMO separation that arises in mathematical chemistry.

In this paper a determinant identity is established, from which a simple proof of the multivariate Lagrange–Good inversion formula follows directly. Further discussion on a discrete analogue of the Lagrange–Good inversion formula is also presented.

In this paper, we study the poset of basic tilting kQ-modules when Q is a Dynkin quiver, and the poset of basic support τ-tilting kQ-modules when Q is a connected acyclic quiver respectively. It is shown that the first poset is a distributive lattice if and only if Q is of types $\mathbb{A}_{1}$, $\mathbb{A}_{2}$ or $\mathbb{A}_{3}$ with a non-linear orientation and the second poset is a distributive lattice if and only if Q is of type $\mathbb{A}_{1}$.

This paper investigates topological reconstruction, related to the reconstruction conjecture in graph theory. We ask whether the homeomorphism types of subspaces of a space X which are obtained by deleting singletons determine X uniquely up to homeomorphism. If the question can be answered affirmatively, such a space is called reconstructible. We prove that in various cases topological properties can be reconstructed. As main result we find that familiar spaces such as the reals ℝ, the rationals ℚ and the irrationals ℙ are reconstructible, as well as spaces occurring as Stone–Čech compactifications. Moreover, some non-reconstructible spaces are discovered, amongst them the Cantor set C.

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.