We show that the maximum number of convex polygons in a triangulation of n points in the plane is O(1.5029n ). This improves an earlier bound of O(1.6181n ) established by van Kreveld, Löffler and Pach (2012), and almost matches the current best lower bound of Ω(1.5028n ) due to the same authors. Given a planar straight-line graph G with n vertices, we also show how to compute efficiently the number of convex polygons in G.

We identify the asymptotic probability of a configuration model CMn (d) producing a connected graph within its critical window for connectivity that is identified by the number of vertices of degree 1 and 2, as well as the expected degree. In this window, the probability that the graph is connected converges to a non-trivial value, and the size of the complement of the giant component weakly converges to a finite random variable. Under a finite second moment condition we also derive the asymptotics of the connectivity probability conditioned on simplicity, from which follows the asymptotic number of simple connected graphs with a prescribed degree sequence.

Let A and B be disjoint sets, of size 2k , of vertices of Qn , the n-dimensional hypercube. In 1997, Bollobás and Leader proved that there must be (n − k)2k edge-disjoint paths between such A and B. They conjectured that when A is a down-set and B is an up-set, these paths may be chosen to be directed (that is, the vertices in the path form a chain). We use a novel type of compression argument to prove stronger versions of these conjectures, namely that the largest number of edge-disjoint paths between a down-set A and an up-set B is the same as the largest number of directed edge-disjoint paths between A and B. Bollobás and Leader made an analogous conjecture for vertex-disjoint paths, and we prove a strengthening of this by similar methods. We also prove similar results for all other sizes of A and B.

A class of graphs is called bridge-addable if, for each graph in the class and each pair u and v of vertices in different components, the graph obtained by adding an edge joining u and v must also be in the class. The concept was introduced in 2005 by McDiarmid, Steger and Welsh, who showed that, for a random graph sampled uniformly from such a class, the probability that it is connected is at least 1/e.

We generalize this and related results to bridge-addable classes with edge-weights which have an edge-expansion property. Here, a graph is sampled with probability proportional to the product of its edge-weights. We obtain for example lower bounds for the probability of connectedness of a graph sampled uniformly from a relatively bridge-addable class of graphs, where some but not necessarily all of the possible bridges are allowed to be introduced. Furthermore, we investigate whether these bounds are tight, and in particular give detailed results about random forests in complete balanced multipartite graphs.

We consider two notions describing how one finite graph may be larger than another. Using them, we prove several theorems for such pairs that compare the number of spanning trees, the return probabilities of random walks, and the number of independent sets, among other combinatorial quantities. Our methods involve inequalities for determinants, for traces of functions of operators, and for entropy.

Computer or communication networks are so designed that they do not easily get disrupted under external attack. Moreover, they are easily reconstructed when they do get disrupted. These desirable properties of networks can be measured by various parameters, such as connectivity, toughness and scattering number. Among these parameters, the isolated scattering number is a comparatively better parameter to measure the vulnerability of networks. In this paper we first prove that for split graphs, this number can be computed in polynomial time. Then we determine the isolated scattering number of the Cartesian product and the Kronecker product of special graphs and special permutation graphs.

Given a family of r-uniform hypergraphs ${\cal F}$ (or r-graphs for brevity), the Turán number ex(n, ${\cal F})$ of ${\cal F}$ is the maximum number of edges in an r-graph on n vertices that does not contain any member of ${\cal F}$ . A pair {u,v} is covered in a hypergraph G if some edge of G contains {u, v}. Given an r-graph F and a positive integer p ⩾ n(F), where n(F) denotes the number of vertices in F, let HFp denote the r-graph obtained as follows. Label the vertices of F as v 1,. . .,vn (F). Add new vertices vn(F)+1 ,. . .,vp . For each pair of vertices vi, vj not covered in F, add a set Bi,j of r − 2 new vertices and the edge {vi, vj } ∪ Bi,j , where the Bi,j are pairwise disjoint over all such pairs {i, j}. We call HFp the expanded p-clique with an embedded F. For a relatively large family of F, we show that for all sufficiently large n, ex(n,HFp ) = |Tr (n, p − 1)|, where Tr (n, p − 1) is the balanced complete (p − 1)-partite r-graph on n vertices. We also establish structural stability of near-extremal graphs. Our results generalize or strengthen several earlier results and provide a class of hypergraphs for which the Turán number is exactly determined (for large n).

Paolo Aluffi, inspired by an algebro-geometric problem, asked when the Kirchhoff polynomial of a graph is in the Jacobian ideal of the Kirchhoff polynomial of the same graph with one edge deleted. We give some results on which graph–edge pairs have this property. In particular, we show that multiple edges can be reduced to double edges, we characterize which edges of wheel graphs satisfy the property, we consider a stronger condition which guarantees the property for any parallel join, and we find a class of series–parallel graphs with the property.

We prove an inequality for functions on the discrete cube {0, 1}n extending the edge-isoperimetric inequality for sets. This inequality turns out to be equivalent to the following claim about random walks on the cube: subcubes maximize ‘mean first exit time’ among all subsets of the cube of the same cardinality.

We investigate the asymptotic version of the Erdős–Ko–Rado theorem for the random k-uniform hypergraph $\mathcal{H}$ k (n, p). For 2⩽k(n) ⩽ n/2, let $N=\binom{n}k$ and $D=\binom{n-k}k$ . We show that with probability tending to 1 as n → ∞, the largest intersecting subhypergraph of $\mathcal{H}$ has size

$$(1+o(1))p\ffrac kn N$$

$$p\gg \ffrac nk\ln^2\biggl(\ffrac nk\biggr)D^{-1}.$$

A different behaviour occurs when k = o(n). In this case, the lower bound on p is almost optimal. Further, for the small interval D −1 ≪ p ⩽ (n/k)1−ϵ D −1, the largest intersecting subhypergraph of $\mathcal{H}$ k (n, p) has size Θ(ln(pD)ND −1), provided that $k \gg \sqrt{n \ln n}$ .

Together with previous work of Balogh, Bohman and Mubayi, these results settle the asymptotic size of the largest intersecting family in $\mathcal{H}$ k , for essentially all values of p and k.

A permutoid is a set of partial permutations that contains the identity and is such that partial compositions, when defined, have at most one extension in the set. In 2004 Peter Cameron conjectured that there can exist no algorithm that determines whether or not a permutoid based on a finite set can be completed to a finite permutation group. In this note we prove Cameron’s conjecture by relating it to our recent work on the profinite triviality problem for finitely presented groups. We also prove that the existence problem for finite developments of rigid pseudogroups is unsolvable. In an appendix, Steinberg recasts these results in terms of inverse semigroups.

The jaggedness of an order ideal $I$ in a poset $P$ is the number of maximal elements in $I$ plus the number of minimal elements of $P$ not in $I$ . A probability distribution on the set of order ideals of $P$ is toggle-symmetric if for every $p\in P$ , the probability that $p$ is maximal in $I$ equals the probability that $p$ is minimal not in $I$ . In this paper, we prove a formula for the expected jaggedness of an order ideal of  $P$ under any toggle-symmetric probability distribution when $P$ is the poset of boxes in a skew Young diagram. Our result extends the main combinatorial theorem of Chan–López–Pflueger–Teixidor [Trans. Amer. Math. Soc., forthcoming. 2015, arXiv:1506.00516], who used an expected jaggedness computation as a key ingredient to prove an algebro-geometric formula; and it has applications to homomesies, in the sense of Propp–Roby, of the antichain cardinality statistic for order ideals in partially ordered sets.

Let $\mathbb{K}$ be a field and S = ${\mathbb{K}}$ [x 1, . . ., xn ] be the polynomial ring in n variables over the field $\mathbb{K}$ . For every monomial ideal I ⊂ S, we provide a recursive formula to determine a lower bound for the Stanley depth of S/I. We use this formula to prove the inequality sdepth(S/I) ≥ size(I) for a particular class of monomial ideals.

In 2005, Knutson–Vakil conjectured a puzzle rule for equivariant $K$ -theory of Grassmannians. We resolve this conjecture. After giving a correction, we establish a modified rule by combinatorially connecting it to the authors’ recently proved tableau rule for the same Schubert calculus problem.

To a pair $P$ and $Q$ of finite posets we attach the toric ring $K[P,Q]$ whose generators are in bijection to the isotone maps from $P$ to $Q$ . This class of algebras, called isotonian, are natural generalizations of the so-called Hibi rings. We determine the Krull dimension of these algebras and for particular classes of posets $P$ and $Q$ we show that $K[P,Q]$ is normal and that their defining ideal admits a quadratic Gröbner basis.

Let $\mathcal F$ ⊂ 2[n] be a family of subsets. The diameter of $\mathcal F$ is the maximum of the size of symmetric differences among pairs of members of $\mathcal F$ . In 1966 Kleitman determined the maximum of | $\mathcal F$ | for fixed diameter. However, this important classical result lacked a characterization of the families meeting the bound. This is remedied in the present paper, where a best possible stability result is established as well.

In Section 4 we introduce a ‘parity trick’ that provides an easy way of deducing the odd case from the even case in both Kleitman's original theorem and its stability version.

In this paper, we give an explicit construction of a quasi-idempotent in the $q$ -rook monoid algebra $R_{n}(q)$ and show that it generates the whole annihilator of the tensor space $U^{\otimes n}$ in $R_{n}(q)$ .

The unconstrained exponential family of random graphs assumes no prior knowledge of the graph before sampling, but it is natural to consider situations where partial information about the graph is known, for example the total number of edges. What does a typical random graph look like, if drawn from an exponential model subject to such constraints? Will there be a similar phase transition phenomenon (as one varies the parameters) as that which occurs in the unconstrained exponential model? We present some general results for this constrained model and then apply them to obtain concrete answers in the edge-triangle model with fixed density of edges.

We consider a marking procedure of the vertices of a tree where each vertex is marked independently from the others with a probability that depends only on its out-degree. We prove that a critical Galton–Watson tree conditioned on having a large number of marked vertices converges in distribution to the associated size-biased tree. We then apply this result to give the limit in distribution of a critical Galton–Watson tree conditioned on having a large number of protected nodes.

In this paper we study the treewidth of the random geometric graph, obtained by dropping n points onto the square [0,√n]2 and connecting pairs of points by an edge if their distance is at most r=r(n). We prove a conjecture of Mitsche and Perarnau (2014) stating that, with probability going to 1 as n→∞, the treewidth of the random geometric graph is 𝜣(r√n) when lim inf r>r c, where r c is the critical radius for the appearance of the giant component. The proof makes use of a comparison to standard bond percolation and with a little bit of extra work we are also able to show that, with probability tending to 1 as k→∞, the treewidth of the graph we obtain by retaining each edge of the k×k grid with probability p is 𝜣(k) if p>½ and 𝜣(√log k) if p<½.