Vizing's conjecture: a survey and recent results

Vizing's conjecture from 1968 asserts that the domination number of the Cartesian product of two graphs is at least as large as the product of their domination numbers. In this paper we survey the approaches to this central conjecture from domination theory and give some new results along the way. For instance, several new properties of a minimal counterexample to the conjecture are obtained and a lower bound for the domination number is proved for products of claw‐free graphs with arbitrary graphs. Open problems, questions and related conjectures are discussed throughout the paper. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 46–76, 2012     

The nature of the magnetism in quasi-2D layered α-Ni(OH)2

The two layered hexagonal hydroxides of Ni are β-Ni(OH)(2) and α-Ni(OH)(2); β-Ni(OH)(2) is now known to be an antiferromagnet whereas the nature of the magnetism in α-Ni(OH)(2) is not yet well established. Here, the magnetic properties of α-Ni(OH)(2) with lattice parameters a = 3.02 ? and c = 8.6 ?, and flower-like morphology with petal thickness of approximately equal to 50 ? are reported. Temperature (2-300 K) and magnetic field (up to 65 kOe) dependence of the magnetization and ac susceptibility at f = 0.1-1000 Hz were measured. Analysis of the data yields ferromagnetic ordering in the system with T(C) is approximately equal to 16 K. In addition, a nanosize related blocking temperature T(B) = 8 K and spin-glass-like ordering of the surface spins near 3.5 K are inferred from the ac frequency and dc magnetic field dependence of these transitions. Fitting to the high temperature series and quasi-2D nature of the system is used to determine J(1)/k(B) = 4.38 K (J(2)/k(B) = 0.14 K) for the intraplane (interplane) exchange coupling between the Ni(2+) ions.     

Solution of finite systems of equations by interval iteration

In actual practice, iteration methods applied to the solution of finite systems of equations yield inconclusive results as to the existence or nonexistence of solutions and the accuracy of any approximate solutions obtained. On the other hand, construction of interval extensions of ordinary iteration operators permits one to carry out interval iteration computationally, with results which can give rigorous guarantees of existence or nonexistence of solutions, and error bounds for approximate solutions. Examples are given of the solution of a nonlinear system of equations and the calculation of eigenvalues and eigenvectors of a matrix by interval iteration. Several ways to obtain lower and upper bounds for eigenvalues are given.Sponsored by the United States Army under Contract No. DAAG29-80-C-0041.     

Distance-Balanced Graphs

Distance-balanced graphs are introduced as graphs in which every edge uv has the following property The number of vertices closer to u than to v is equal to the number of vertices closer to v than to u. Basic properties of these graphs are obtained. The new concept is connected with symmetry conditions in graphs and local operations on graphs are studied with respect to it. Distance-balanced Cartesian and lexicographic products of graphs are also characterized. Several open problems are posed along the way. Received August 31, 2005     

Influence of thickness,width and temperature on critical current density of Nb thin film structures

The density of critical currents j_C in Nb thin films with thickness smaller than 15 nm and width between 100 nm and 10 μm has been measured in a wide temperature range. We have found that the temperature dependencies of j_C in sub-micrometer wide bridges at 0.7T_C < T < T_C are well described by the Ginzburg–Landau de-pairing critical current. In wider bridges already at T < 0.9T_C the j_C value is significantly reduced due to the penetration and de-pinning of magnetic vortices.