On a cycle partition problem

Let G be any graph and let c(G) denote the circumference of G. We conjecture that for every pair c₁,c₂ of positive integers satisfying c₁+c₂=c(G), the vertex set of G admits a partition into two sets V₁ and V₂, such that V_i induces a graph of circumference at most c_i, i=1,2. We establish various results in support of the conjecture; e.g. it is observed that planar graphs, claw-free graphs, certain important classes of perfect graphs, and graphs without too many intersecting long cycles, satisfy the conjecture.This work is inspired by a well-known, long-standing, analogous conjecture involving paths.     

Property A and affine buildings

Yu's Property A is a non-equivariant generalisation of amenability introduced in his study of the coarse Baum Connes conjecture. In this paper we show that all affine buildings of type , and have Property A. Together with results of Guentner, Higson and Weinberger, this completes a programme to show that all affine building have Property A. In passing we use our technique to obtain a new proof for groups acting on buildings.     

A note on the Path Kernel Conjecture

Let τ(G) denote the number of vertices in a longest path in a graph G=(V,E). A subset K of V is called a P_n-kernel of G if τ(G[K])≤n−1 and every vertex v∈V?K is adjacent to an end-vertex of a path of order n−1 in G[K]. It is known that every graph has a P_n-kernel for every positive integer n≤9. R. Aldred and C. Thomassen in [R.E.L. Aldred, C. Thomassen, Graphs with not all possible path-kernels, Discrete Math. 285 (2004) 297-300] proved that there exists a graph which contains no P₃₆₄-kernel. In this paper, we generalise this result. We construct a graph with no P₁₅₅-kernel and for each integer l≥0 we provide a construction of a graph G containing no P_τ(G)−l-kernel.     

The coarse Baum–Connes conjecture via coarse geometry

The C₀ coarse structure on a metric space is a refinement of the bounded structure and is closely related to the topology of the space. In this paper we will prove the C₀ version of the coarse Baum–Connes conjecture and show that K_*(C^*X₀) is a topological invariant for a broad class of metric spaces. Using this result we construct a ‘geometric’ obstruction group to the coarse Baum–Connes conjecture for the bounded coarse structure. We then show under the assumption of finite asymptotic dimension that the obstructions vanish, and hence we obtain a new proof of the coarse Baum–Connes conjecture in this context.     

On the Total Domination Number of Cartesian Products of Graphs

The most famous open problem involving domination in graphs is Vizings conjecture which states the domination number of the Cartesian product of any two graphs is at least as large as the product of their domination numbers. In this paper, we investigate a similar problem for total domination. In particular, we prove that the product of the total domination numbers of any nontrivial tree and any graph without isolated vertices is at most twice the total domination number of their Cartesian product, and we characterize the extremal graphs.Research supported in part by the South African National Research Foundation and the University of KwaZulu-Natal     

On the Bergman representative coordinates

We study the set where the so-called Bergman representative coordinates (or Bergman functions) form an immersion. We provide an estimate of the size of a maximal geodesic ball with respect to the Bergman metric contained in this set. By concrete examples we show that these estimates are the best possible.     

The evolution of invariant manifolds in Hamiltonian-Hopf bifurcations

We consider the evolution of the stable and unstable manifolds of an equilibrium point of a Hamiltonian system of two degrees of freedom which depends on a parameter, ν. The eigenvalues of the linearized system are complex for ν<0 and pure imaginary for ν>0. Thus, for ν<0 the equilibrium has a two-dimensional stable manifold and a two-dimensional unstable manifold, but for ν>0 these stable and unstable manifolds are gone. If the sign of a certain term in the normal form is positive then for small negative ν the stable and unstable manifolds of the system are either identical or must have transverse intersection. Thus, either the system is totally degenerate or the system admits a suspended Smale horseshoe as an invariant set.     

An illustrated theory of hyperbolic virtual polytopes

The paper gives an illustrated introduction to the theory of hyperbolic virtual polytopes and related counterexamples to A.D. Alexandrov’s conjecture.