The chromatic number and other functions of the lexicographic product

The values of the chromatic and achromatic number, point- and line-connectivity, and point independence number of the lexicographic product of two graphs are examined in relation to the values of the respective parameters on the factor graphs.     

A combinatorial analog of the Jordan Curve Theorem

The concept of the genus of a pair of permutations is defined in the same manner as was done by Jacques. The integrality of the genus is proven in a new way by applying a technique developed by Walkup for the reduction of products of permutations. These tools are then used to prove an analog of the Jordan Curve Theorem for a pair of permutations whose genus is zero. A Jordan Curve Theorem for plane embeddings of graphs then follows as a corollary.     

A possible force between three nucleons due to the self-coupling of three gluons

We estimate the strength of a new three-body force between three nucleons in close proximity which arises from the self-coupling of three coloured gluons. It is possible that this force can account for the recently measured depression of the central charge density in ${}^3\text{He}$.     

Generalized Embedding Schemes

Covering projections are used to provide a logical foundation for the generalized embedding schemes which describe embeddings of graphs on closed surfaces that are not necessarily orientable. Criteria are provided to resolve the question of when such schemes describe the same embedding, and when that embedding is orientable. The schemes are then applied to derive some interpolation-type theorems for embeddings on closed surfaces.     

Influence of architecture on the kinetic stability of molecular assemblies

The strength of a multimolecular system depends on the number of interactions that hold it together. Using dynamic force spectroscopy, we show how the kinetic stability of a system decreases as the number of molecular bonds is increased, as predicted by theory. The data raise important considerations for experimental tests of bond strength and, as a paradigm, suggest both routes to and pitfalls in methods for computational simulation of molecular transitions, such as ligand binding and protein folding.     

Toward a best adiabatic potential

The concept of a best adiabatic potential is discussed. It is emphasized that it should be best for a number of states, not just for a single state. One particular figure of merit is introduced and applied to a simple model.     

Gas phase acidities of CH bonds adjacent to oxygen and to sulphur

It has been shown, by double zeta quality ab initio MO calculations, that, in the gas phase, a CH bond adjacent to sulphur is more acidic than a CH bond adjacent to oxygen. This trend is in agreement with experimental observations in solution.     

n-Tuple colorings and associated graphs

The n-tuple graph coloring, which assigns to each vertex n colors, is defined together with its respective chromatic number x_n. It is proved that these numbers satisfy the inequality x_n ≥ 2 + x_n?1, and that equality holds only for bipartite graphs. Graphs $\text{G}{}_{\mathrm{n}}{}^{\mathrm{m}}$ are defined which play the same role for the n-tuple coloring that K_m plays for the conventional coloring. The chromatic numbers of various classes of graphs are also calculated.