Let G be a graph with n vertices, and let k be an integer dividing n. G is said to be strongly k-colourable if, for every partition of V(G) into disjoint sets V1 ∪ ··· ∪ Vr, all of size exactly k, there exists a proper vertex k-colouring of G with each colour appearing exactly once in each Vi. In the case when k does not divide n, G is defined to be strongly k-colourable if the graph obtained by adding isolated vertices is strongly k-colourable. The strong chromatic number of G is the minimum k for which G is strongly k-colourable. In this paper, we study the behaviour of this parameter for the random graph Gn,p. In the dense case when p ≫ n−1/3, we prove that the strong chromatic number is a.s. concentrated on one value Δ + 1, where Δ is the maximum degree of the graph. We also obtain several weaker results for sparse random graphs.

Semi-graphoids are combinatorial structures that arise in statistical learning theory. They are equivalent to convex rank tests and to polyhedral fans that coarsen the reflection arrangement of the symmetric group Sn. In this paper we resolve two problems on semi-graphoids posed in Studený's book (2005), and we answer a related question of Postnikov, Reiner and Williams on generalized permutohedra. We also study the semigroup and the toric ideal associated with semi-graphoids.

A proper vertex colouring of a graph is equitable if the sizes of colour classes differ by at most one. We present a new shorter proof of the celebrated Hajnal–Szemerédi theorem: for every positive integer r, every graph with maximum degree at most r has an equitable colouring with r+1 colours. The proof yields a polynomial time algorithm for such colourings.

We prove tail estimates for variables of the form ∑if(Xi), where (Xi)i is a sequence of states drawn from a reversible Markov chain, or, equivalently, from a random walk on an undirected graph. The estimates are in terms of the range of the function f, its variance, and the spectrum of the graph. The purpose of our estimates is to determine the number of chain/walk samples which are required for approximating the expectation of a distribution on vertices of a graph, especially an expander. The estimates must therefore provide information for fixed number of samples (as in Gillman's [4]) rather than just asymptotic information. Our proofs are more elementary than other proofs in the literature, and our results are sharper. We obtain Bernstein- and Bennett-type inequalities, as well as an inequality for sub-Gaussian variables.

We prove that the chromatic polynomial of a finite graph of maximal degree Δ is free of zeros for |q| ≥ C*(Δ) withThis improves results by Sokal and Borgs. Furthermore, we present a strengthening of this condition for graphs with no triangle-free vertices.

We consider the number of vertices that must be removed from a graph G in order that the remaining subgraph have no component with more than k vertices. Our principal observation is that, if G is a sparse random graph or a random regular graph on n vertices with n → ∞, then the number in question is essentially the same for all values of k that satisfy both k → ∞ and k =o(n).

Given m positive integers R = (ri), n positive integers C = (cj) such that Σri = Σcj = N, and mn non-negative weights W=(wij), we consider the total weight T=T(R, C; W) of non-negative integer matrices D=(dij) with the row sums ri, column sums cj, and the weight of D equal to . For different choices of R, C, and W, the quantity T(R,C; W) specializes to the permanent of a matrix, the number of contingency tables with prescribed margins, and the number of integer feasible flows in a network. We present a randomized algorithm whose complexity is polynomial in N and which computes a number T′=T′(R,C;W) such that T′ ≤ T ≤ α(R,C)T′ where . In many cases, ln T′ provides an asymptotically accurate estimate of ln T. The idea of the algorithm is to express T as the expectation of the permanent of an N × N random matrix with exponentially distributed entries and approximate the expectation by the integral T′ of an efficiently computable log-concave function on ℝmn.

Let Γ =(V,E) be a point-symmetric reflexive relation and let υ ∈ V such that |Γ(υ)| is finite (and hence |Γ(x)| is finite for all x, by the transitive action of the group of automorphisms). Let j ∈ℕ be an integer such that Γj(υ)∩ Γ−(υ)={υ}. Our main result states that

As an application we have |Γj(υ)| ≥ 1+(|Γ(υ)|−1)j. The last result confirms a recent conjecture of Seymour in the case of vertex-symmetric graphs. Also it gives a short proof for the validity of the Caccetta–Häggkvist conjecture for vertex-symmetric graphs and generalizes an additive result of Shepherdson.

Zeilberger's enumeration schemes can be used to completely automate the enumeration of many permutation classes. We extend his enumeration schemes so that they apply to many more permutation classes and describe the Maple package WilfPlus, which implements this process. We also compare enumeration schemes to three other systematic enumeration techniques: generating trees, substitution decompositions, and the insertion encoding.

We consider random graphs with a fixed degree sequence. Molloy and Reed [11, 12] studied how the size of the giant component changes according to degree conditions. They showed that there is a phase transition and investigated the order of components before and after the critical phase. In this paper we study more closely the order of components at the critical phase, using singularity analysis of a generating function for a branching process which models the random graph with a given degree sequence.

Let denote the set of unrooted labelled trees of size n and let ℳ be a particular (finite, unlabelled) tree. Assuming that every tree of is equally likely, it is shown that the limiting distribution as n goes to infinity of the number of occurrences of ℳ is asymptotically normal with mean value and variance asymptotically equivalent to μn and σ2n, respectively, where the constants μ>0 and σ≥0 are computable.

The k-core of a graph G is the maximal subgraph of G having minimum degree at least k. In 1996, Pittel, Spencer and Wormald found the threshold λc for the emergence of a non-trivial k-core in the random graph G(n, λ/n), and the asymptotic size of the k-core above the threshold. We give a new proof of this result using a local coupling of the graph to a suitable branching process. This proof extends to a general model of inhomogeneous random graphs with independence between the edges. As an example, we study the k-core in a certain power-law or ‘scale-free’ graph with a parameter c controlling the overall density of edges. For each k ≥ 3, we find the threshold value of c at which the k-core emerges, and the fraction of vertices in the k-core when c is ϵ above the threshold. In contrast to G(n, λ/n), this fraction tends to 0 as ϵ→0.

In a balls-in-bins process with feedback, balls are sequentially thrown into bins so that the probability that a bin with n balls obtains the next ball is proportional to f(n) for some function f. A commonly studied case where there are two bins and f(n) = np for p > 0, and our goal is to study the fine behaviour of this process with two bins and a large initial number t of balls. Perhaps surprisingly, Brownian Motions are an essential part of both our proofs.

For p > 1/2, it was known that with probability 1 one of the bins will lead the process at all large enough times. We show that if the first bin starts with balls (for constant λ∈ℝ), the probability that it always or eventually leads has a non-trivial limit depending on λ.

For p ≤ 1/2, it was known that with probability 1 the bins will alternate in leadership. We show, however, that if the initial fraction of balls in one of the bins is > 1/2, the time until it is overtaken by the remaining bin scales like Θ(t1+1/(1-2p)) for p < 1/2 and exp(Θ(t)) for p = 1/2. In fact, the overtaking time has a non-trivial distribution around the scaling factor, which we determine explicitly.

Our proofs use a continuous-time embedding of the balls-in-bins process (due to Rubin) and a non-standard approximation of the process by Brownian Motion. The techniques presented also extend to more general functions f.

This paper treats a simple model, which can be exactly solved, motivated by the back-and-forth motion of ocean bacteria. In particular, the probability is determined that a bacterium moving randomly along a fluid line through the origin in a linear shear flow hits the origin before time t.

In this article, a discrete mean value of the derivative of the Riemann zeta function is computed. This mean value will be important for several applications concerning the size of ζ′(ρ), where ζ(s) is the Riemann zeta function and ρ is a non-trivial zero of ζ(s).

Let be a family of axis-aligned parallelotopes, or boxes, in ℝd. Denote by fk () the number of subfamilies of of size k + 1 with non-empty intersection. In an earlier paper, the author proved that, if f0 () = n and fr() = 0, then fk() ≤ fk(n, d, r) for k = 1,…,r − 1, where fk(n, d, r) is some explicitly given number. The result is best possible for all k. Here it is shown that, if equality is attained for some such k, then equality is attained for each such k.

A Minkowski class is a closed subset of the space of convex bodies in Euclidean space ℝn which is closed under Minkowski addition and non-negative dilatations. A convex body in ℝn is universal if the expansion of its support function in spherical harmonics contains non-zero harmonics of all orders. If K is universal, then a dense class of convex bodies M has the following property. There exist convex bodies T1, T2 such that M + T1 = T2, and T1, T2 belong to the rotation invariant Minkowski class generated by K. It is shown that every convex body K which is not centrally symmetric has a linear image, arbitrarily close to K, which is universal. A modified version of the result holds for centrally symmetric convex bodies. In this way, a result of S. Alesker is strengthened, and at the same time given a more elementary proof.

Variations and generalizations of several classical theorems concerning characterizations of ellipsoids are developed. In particular, these lead to a short and comprehensible proof of the false centre theorem.

Let X ⊂ ℙN be a geometrically integral cubic hypersurface defined over ℚ, with singular locus of dimension at most dim X − 4. The main result in this paper is a proof of the fact that X(ℚ) contains OɛX (BdimX + ɛ) points of height at most B.

Let K be a convex body of dimension at least 3, and let p0 be a point. If every section of K through p0 is centrally symmetric, then Rogers proved in [6] that K is centrally symmetric, although p0 may not be the centre of K. If this is the case, then Aitchison, Petty and Rogers [1] and Larman [2] proved that K must be an ellipsoid. Suppose now that, for every direction, we can choose continuously a section of K that is centrally symmetric; if K is strictly convex, then Montejano [3] proved that K must be centrally symmetric. Consider now the following example. Let D be a (euclidean) ball centred at the origin from which two symmetric caps are deleted. Then D is centrally symmetric with respect to the origin, and has a lot of circular sections whose centre is not the origin. In fact, we can choose continuously, for every direction, a section of D which is centrally symmetric, in such a way that not all these sections pass through the origin. Nevertheless, no matter how we choose these sections, there are always necessarily many of them that do pass through the origin. For those sections, of course, we have not imposed any condition, which explains the fact that D is not a quadric elsewhere.