A graph is s-transitive if its automorphism group acts transitively on s-arcs but not on (s+1)-arcs in the graph. Let X be a connected tetravalent s-transitive graph of order twice a prime power. In this paper it is shown that s=1,2,3 or 4. Furthermore, if s=2, then X is a normal cover of one of the following graphs: the 4-cube, the complete graph of order 5, the complete bipartite graph K5,5 minus a 1-factor, or K7,7 minus a point-hyperplane incidence graph of the three-dimensional projective geometry PG(2,2); if s=3, then X is a normal cover of the complete bipartite graph of order 4; if s=4, then X is a normal cover of the point-hyperplane incidence graph of the three-dimensional projective geometry PG(2,3). As an application, we classify the tetravalent s-transitive graphs of order 2p2 for prime p.

We present new ideas about the type of random tessellation which evolves through successive division of its cells. These ideas are developed in an intuitive way, with many pictures and only a modicum of mathematical formalism–so that the wide application of the ideas is clearly apparent to all readers. A vast number of new tessellation models, with known probability distribution for the volume of the typical cell, follow from the concepts in this paper. There are other interesting models for which results are not presented (or presented only through simulation methods), but these models have illustrative value. A large agenda of further research is opened up by the ideas in this paper.

We consider a variety of subtrees of various shapes lying on the fringe of a recursive tree. We prove that (under suitable normalization) the number of isomorphic images of a given fixed tree shape on the fringe of the recursive tree is asymptotically Gaussian. The parameters of the asymptotic normal distribution involve the shape functional of the given tree. The proof uses the contraction method.

We consider a multicommodity flow problem on a complete graph whose edges have random, independent, and identically distributed capacities. We show that, as the number of nodes tends to infinity, the maximum utility, given by the average of a concave function of each commodity flow, has an almost-sure limit. Furthermore, the asymptotically optimal flow uses only direct and two-hop paths, and can be obtained in a distributed manner.

Let be a Poisson process of intensity one in the infinite plane ℝ2. We surround each point x of by the open disc of radius r centred at x. Now let S n be a fixed disc of area n, and let C r (S n ) be the set of discs which intersect S n . Write E r k for the event that C r (S n ) is a k-cover of S n , and F r k for the event that C r (S n ) may be partitioned into k disjoint single covers of S n . We prove that P(E r k ∖ F r k ) ≤ c k / logn, and that this result is best possible. We also give improved estimates for P(E r k ). Finally, we study the obstructions to k-partitionability in more detail. As part of this study, we prove a classification theorem for (deterministic) covers of ℝ2 with half-planes that cannot be partitioned into two single covers.

We discuss n4 configurations of n points and n planes in three-dimensional projective space. These have four points on each plane, and four planes through each point. When the last of the 4n incidences between points and planes happens as a consequence of the preceding 4n−1 the configuration is called a ‘theorem’. Using a graph-theoretic search algorithm we find that there are two 84 and one 94 ‘theorems’. One of these 84 ‘theorems’ was already found by Möbius in 1828, while the 94 ‘theorem’ is related to Desargues’ ten-point configuration. We prove these ‘theorems’ by various methods, and connect them with other questions, such as forbidden minors in graph theory, and sets of electrons that are energy minimal.

We construct the Schubert basis of the torus-equivariant K-homology of the affine Grassmannian of a simple algebraic group G, using the K-theoretic NilHecke ring of Kostant and Kumar. This is the K-theoretic analogue of a construction of Peterson in equivariant homology. For the case where G=SLn, the K-homology of the affine Grassmannian is identified with a sub-Hopf algebra of the ring of symmetric functions. The Schubert basis is represented by inhomogeneous symmetric functions, calledK-k-Schur functions, whose highest-degree term is a k-Schur function. The dual basis in K-cohomology is given by the affine stable Grothendieck polynomials, verifying a conjecture of Lam. In addition, we give a Pieri rule in K-homology. Many of our constructions have geometric interpretations by means of Kashiwara’s thick affine flag manifold.

Let G be a graph, and k a positive integer. Let h:E(G)→[0,1] be a function. If ∑ e∋xh(e)=k holds for each x∈V (G), then we call G[Fh] a fractional k-factor of G with indicator function h where Fh={e∈E(G)∣h(e)>0}. In this paper we use neighbourhoods to obtain a new sufficient condition for a graph to have a fractional k-factor. Furthermore, this result is shown to be best possible in some sense.

In this paper, compositions of a natural number are studied. The number of restricted compositions is given in a closed form, and some applications are presented.

For a class of ‘linear’ sudoku solutions, we construct mutually orthogonal families of maximal size for all square orders, and we show that all such solutions must lie in the same orbit of a symmetry group preserving sudoku solutions.

Colmez has given a recipe to associate a smooth modular representation Ω(W) of the Borel subgroup of GL2(Qp) to a -representation W of by using Fontaine’s theory of (φ,Γ)-modules. We compute Ω(W) explicitly and we prove that if W is irreducible and dim (W)=2, then Ω(W) is the restriction to the Borel subgroup of GL2(Qp) of the supersingular representation associated to W by Breuil’s correspondence.

We prove that for any given integer c≥0 any metric space on n points may be isometrically embedded into ln−c∞ provided n is large enough.

It is well known that a tournament (complete oriented graph) on n vertices has at most directed triangles, and that the constant is best possible. Motivated by some geometric considerations, our aim in this paper is to consider some “higher order” versions of this statement. For example, if we give each 3-set from an n-set a cyclic ordering, then what is the greatest number of “directed 4-sets” we can have? We give an asymptotically best possible answer to this question, and give bounds in the general case when we orient each d-set from an n-set.

We prove that there are permutation classes (hereditary properties of permutations) of every growth rate (Stanley–Wilf limit) at least λ≈2.48187, the unique real root of x5−2x4−2x2−2x−1, thereby establishing a conjecture of Albert and Linton.

Let K n denote the number of types of a sample of size n taken from an exchangeable coalescent process (Ξ-coalescent) with mutation. A distributional recursion for the sequence (K n )n∈ℕ is derived. If the coalescent does not have proper frequencies, i.e. if the characterizing measure Ξ on the infinite simplex Δ does not have mass at 0 and satisfies ∫Δ ∣x∣Ξ(dx)/(x,x)<∞, where ∣x∣:=∑i=1 ∞x i and (x,x)≔∑i=1 ∞x i 2 for x=(x 1,x 2,…)∈Δ, then K n /n converges weakly as n→∞ to a limiting variable K that is characterized by an exponential integral of the subordinator associated with the coalescent process. For so-called simple measures Ξ satisfying ∫ΔΞ(d x)/(x,x)<∞, we characterize the distribution of K via a fixed-point equation.

In this paper we focus on the problem of the degree sequence for a random graph process with edge deletion. We prove that, while a specific parameter varies, the limit degree distribution of the model exhibits critical phenomenon.

The probability that two randomly selected phylogenetic trees of the same size are isomorphic is found to be asymptotic to a decreasing exponential modulated by a polynomial factor. The number of symmetrical nodes in a random phylogenetic tree of large size obeys a limiting Gaussian distribution, in the sense of both central and local limits. The probability that two random phylogenetic trees have the same number of symmetries asymptotically obeys an inverse square-root law. Precise estimates for these problems are obtained by methods of analytic combinatorics, involving bivariate generating functions, singularity analysis, and quasi-powers approximations.

We situate the noncrossing partitions associated with a finite Coxeter group within the context of the representation theory of quivers. We describe Reading’s bijection between noncrossing partitions and clusters in this context, and show that it extends to the extended Dynkin case. Our setup also yields a new proof that the noncrossing partitions associated with a finite Coxeter group form a lattice. We also prove some new results within the theory of quiver representations. We show that the finitely generated, exact abelian, and extension-closed subcategories of the representations of a quiver Q without oriented cycles are in natural bijection with the cluster tilting objects in the associated cluster category. We also show that these subcategories are exactly the finitely generated categories that can be obtained as the semistable objects with respect to some stability condition.

For three points , and in the n-dimensional space 𝔽nq over the finite field 𝔽q of q elements we give a natural interpretation of an acute angle triangle defined by these points. We obtain an upper bound on the size of a set 𝒵 such that all triples of distinct points define acute angle triangles. A similar question in the real space ℛn dates back to P. Erdős and has been studied by several authors.