Switchings, extensions, and reductions in central digraphs

A directed graph is called central if its adjacency matrix A satisfies the equation A²=J, where J is the matrix with a 1 in each entry. It has been conjectured that every central directed graph can be obtained from a standard example by a sequence of simple operations called switchings, and also that it can be obtained from a smaller one by an extension. We disprove these conjectures and present a general extension result which, in particular, shows that each counterexample extends to an infinite family.     

On planarity of compact, locally connected, metric spaces

Independently, Claytor [Ann. Math. 35 (1934), 809–835] and Thomassen [Combinatorica 24 (2004), 699–718] proved that a 2-connected, compact, locally connected metric space is homeomorphic to a subset of the sphere if and only if it does not contain K ₅ or K _3;3. The “thumbtack space” consisting of a disc plus an arc attaching just at the centre of the disc shows the assumption of 2-connectedness cannot be dropped. In this work, we introduce “generalized thumbtacks” and show that a compact, locally connected metric space is homeomorphic to a subset of the sphere if and only if it does not contain K ₅, K _3;3, or any generalized thumbtack, or the disjoint union of a sphere and a point.     

Distances in orientations of graphs

We prove that there is a function h(k) such that every undirected graph G admits an orientation H with the following property: If an edge uv belongs to a cycle of length k in G, then uv or vu belongs to a directed cycle of length at most h(k) in H. Next, we show that every undirected bridgeless graph of radius r admits an orientation of radius at most r² + r, and this bound is best possible. We consider the same problem with radius replaced by diameter. Finilly, we show that the problem of deciding whether an undirected graph admits an orientation of diameter (resp. radius) 2 belongs to a class of problems called NP-hard.     

Finite metric spaces of strictly negative type

We prove that, if a finite metric space is of strictly negative type, then its transfinite diameter is uniquely realized by the infinite extender (load vector). Finite metric spaces that have this property include all spaces on two, three, or four points, all trees, and all finite subspaces of Euclidean spaces. We prove that, if the distance matrix is both hypermetric and regular, then it is of strictly negative type. We show that the strictly negative type finite subspaces of spheres are precisely those which do not contain two pairs of antipodal points. In connection with an open problem raised by Kelly, we conjecture that all finite subspaces of hyperbolic spaces are hypermetric and regular, and hence of strictly negative type.     

A nine point theorem for 3-connected graphs

We prove that a 3-connected cubic graph contains a cycle through any nine points.     

A short proof of Meyniel's theorem

Meyniel's theorem states that a strict diconnected digraph has a directed Hamilton cycle if d(u) + d(v) ? 2n ? 1 for every pair u, v of nonadjacent vertices. We give short proof of this theorem.     

Counterexamples to faudree and schelp's conjecture on hamiltonian-connected graphs

Faudree and Schelp conjectured that for any two vertices x, y in a Hamiltonian-connected graph G and for any integer k, where n/2 ? k ? n ? 1, G has a path of length k connecting x and y. However, we show in this paper that there are infinitely many exceptions to this conjecture and we comment on some problems on path length distribution raised by Faudree and Schelp.     

Spectral interferences from phosphate matrices in the determination of arsenic, antimony, selenium and tellurium by electrothermal atomic absorption spectrometry

The spectral interferences of phosphorus species originating from the thermal decomposition of calcium phosphate on Sb, As, Se and Te resonance lines, and the influence of increasing amounts of Ce, Ni, W, Pd, Pt, Zr and other elements on non-correctable signals generated by calcium phosphate and on selenium and phosphorus sensitivity have been studied. The results indicate that spectral interference is caused by P₂ absorption and that the extent of interference depends on the wavelength and the spectral band width. The generation of P₂ is masked by large amounts of all the tested reagents. There is a significant reduction in selenium sensitivity in the presence of high concentrations of Ce, Pd and Pt while no decrease in sensitivity is caused by the presence of even 1% nickel and tungsten solutions. All the reagents tested provided enhanced phosphorus sensitivity, thorium being the best.