Extending an edge-coloring

When can a k-edge-coloring of a subgraph K of a graph G be extended to a k-edge-coloring of G? One necessary condition is that for all X ? E(G) - E(K), where μ_i(X) is the maximum cardinality of a subset of X whose union with the set of edges of K colored i is a matching. This condition is not sufficient in general, but is sufficient for graphs of very simple structure. We try to locate the border where sufficiency ends.     

A note on growing binary trees

The class ? of binary search trees is studied. A leaf is a vertex of degree 0; ?_n is the subset of ? consisting of trees with n leaves. We grow trees in ?_n from ?_n ? 1 thereby inducing a probability measure on ?_n. We will show that the expected value of the average leaf distance of t ∈ ?_n is asymptotic to log₂n as n → ∞.     

Quasi-localized representation of electronic band structures in cubic crystals

Tabulated results and formulae are given for the parametrization of electronic band structures in simple cubic, body-centered cubic and face-centred cubic systems. This technique is based on the decoupling transformation and does not involve least squares fitting.A band labelling scheme is introduced which allows the decoupling transformation to be used to define the energy matrix of Wannier states in copper. This approach is used to re-examine the significance of the tight binding approximation for the d-states and to relate Fermi surface parameters directly to calculated band energies.     

On the calculation of charge densities in crystals

A general method is given for determining the spatial charge distribution of a filled electronic band using Bloch state charge distributions determined at very few k points. Formulae are given for several important cases.