We present some nice properties of the classical construction of triangle-free graphs with high chromatic number given by Blanche Descartes and its modifications. In particular, we construct colour-critical graphs and hypergraphs of high girth with moderate average degree.

Consider a finite alphabet Ω and strings consisting of elements from Ω. For a given string w, let cor(w) denote the autocorrelation, which can be seen as a measure of the amount of overlap in w. Furthermore, let aw(n) be the number of strings of length n that do not contain w as a substring. Eriksson [4] stated the following conjecture: if cor(w)>cor(w′), thenaw(n)>aw′(n) from the first n where equality no longer holds. We prove that this is true if [mid ]Ω[mid ][ges ]3, by giving a lower bound for aw(n)−aw′(n).

Let C(G) denote the number of simple cycles of a graph G and let C(n) be the maximum of C(G) over all planar graphs with n nodes. We present a lower bound on C(n), constructing graphs with at least 2.28n cycles. Applying some probabilistic arguments we prove an upper bound of 3.37n.

We also discuss this question restricted to the subclasses of grid graphs, bipartite graphs, and 3-colourable triangulated graphs.

We show that the Poisson–Dirichlet distribution is the distribution of points in a scale-invariant Poisson process, conditioned on the event that the sum T of the locations of the points in (0,1] is 1. This extends to a similar result, rescaling the locations by T, and conditioning on the event that T[les ]1. Restricting both processes to (0, β] for 0<β[les ]1, we give an explicit formula for the total variation distance between their distributions. Connections between various representations of the Poisson–Dirichlet process are discussed.

A graph G is m-choosable with impropriety d, or simply (m, d)*-choosable, if, for every list assignment L, where [mid ]L(v)[mid ][ges ]m for every v∈V(G), there exists an L-colouring of G such that each vertex of G has at most d neighbours coloured with the same colour as itself. We prove a Grötzsch-type theorem for list colourings with impropriety one, that is, the (3, 1)*-choosability for triangle-free planar graphs; in the proof the method of extending a precolouring of a 4- or 5-cycle is used.

We introduce the path resistance method for lower bounds on the smallest nontrivial eigenvalue of the Laplacian matrix of a graph. The method is based on viewing the graph in terms of electrical circuits: it uses clique embeddings to produce lower bounds on λ2 and star embeddings to produce lower bounds on the smallest Rayleigh quotient when there is a zero Dirichlet boundary condition. The method assigns priorities to the paths in the embedding; we show that, for an unweighted tree T, using uniform priorities for a clique embedding produces a lower bound on λ2 that is off by at most an O(log diameter(T)) factor. We show that the best bounds this method can produce for clique embeddings are the same as for a related method that uses clique embeddings and edge lengths to produce bounds.

We consider the problem of determining the maximum number N(m, k, r) of columns of a 0−1 matrix with m rows and exactly r ones in each column such that every k columns are linearly independent over ℤ2. For fixed integers k[ges ]4 and r[ges ]2, where k is even and gcd(k−1, r) = 1, we prove the lower bound N(m, k, r) = Ω(mkr/2(k−1)·(ln m)1/k−1). This improves on earlier results from [14] by a factor Θ((ln m)1/k−1). Moreover, we describe a polynomial time algorithm achieving this new lower bound.

Let A be a subset of an abelian group G. The subset sum of A is the set [sum ](A) = {[sum ]x∈T[mid ]T⊂A}. We prove the following result. Let S be a generating subset of an abelian group G such that 0∉S and 14[les ][mid ]S[mid ]. Then one of the following conditions holds.

(i) [mid ][sum ](S)[mid ][ges ]min([mid ]G[mid ]−3, 3[mid ]S[mid ]−3).

(ii) There is an x∈S such that S[setmn ]{x} generates a proper subgroup of order less than (3[mid ]S[mid ]−3)/2.

As a consequence, we obtain the following open case of an old conjecture of Diderrich. Let q be a composite odd number and let G be an abelian group of order 3q. Let S be a subset of G with cardinality q+1. Then every element of G is the sum of some subset of S.

Let [Fscr] be a family of forbidden k-hypergraphs (k-uniform set systems). An [Fscr]-saturated hypergraph is a maximal k-uniform set system not containing any member of [Fscr]. As the main result we prove that, for any finite family [Fscr], the minimum number of edges of an [Fscr]-saturated hypergraph is O(nk−1). In particular, this implies a conjecture of Tuza. Some other related results are presented.

Using differential equations, we examine the GREEDY algorithm studied by Azar, Broder, Karlin and Upfal for distributed load balancing [1]. This approach yields accurate estimates of the actual load distribution, provides insight into the exponential improvement GREEDY offers over simple random selection, and allows one to prove tight concentration theorems about the loads in a straightforward manner.

As a paradigm for non-interpenetrating crack models, the Poisson equation in a nonsmooth domain in R2 is considered. The geometrical domain has a cut (a crack) of variable length. At the crack faces, inequality type boundary conditions are prescribed. The behaviour of the energy functional is analysed with respect to the crack length changes. In particular, the derivative of the energy functional with respect to the crack length is obtained. The associated Griffith formula is derived, and properties of the solution are investigated. It is shown that the Rice–Cherepanov integral defined for the solutions of the unilateral problem defined in the nonsmooth domain is path-independent. Finally, a non-negative measure characterising interaction forces between the crack faces is constructed.

We study a mean-field model of superconducting vortices in one and two dimensions. The existence of a weak solution and a steady-state solution of the model are proved. A special case of the steady-state problem is shown to be of the form of a free boundary problem. The solutions of this free boundary problem are investigated. It is also shown that the weak solution of the one-dimensional model is unique and satisfies an entropy inequality.

The thermal equilibrium state of a bipolar, isothermal quantum fluid confined to a bounded domain Ω⊂ℝd, d = 1, 2 or d = 3 is the minimizer of the total energy [Escr]ελ; [Escr]ελ involves the squares of the scaled Planck's constant ε and the scaled minimal Debye length λ. In applications one frequently has λ2[Lt]1. In these cases the zero-space-charge approximation is rigorously justified. As λ → 0, the particle densities converge to the minimizer of a limiting quantum zero-space-charge functional exactly in those cases where the doping profile satisfies some compatibility conditions. Under natural additional assumptions on the internal energies one gets an differential-algebraic system for the limiting (λ = 0) particle densities, namely the quantum zero-space-charge model. The analysis of the subsequent limit ε → 0 exhibits the importance of quantum gaps. The semiclassical zero-space-charge model is, for small ε, a reasonable approximation of the quantum model if and only if the quantum gap vanishes. The simultaneous limit ε = λ → 0 is analyzed.

We study the interior spike solutions to a steady state problem of the shadow system of the Gierer–Meinhardt system arising from biological pattern formation. We first show that at a non-degenerate peak point the interior spike solution is locally unique, and then we establish the spectrum estimates of the associated linearized operator. We also prove that the corresponding solution to the shadow system is unstable. Furthermore, the metastability of such solutions is analysed.