Pagenumber of complete bipartite graphs

Given an ordering of the vertices of a graph around a circle, a page is a collection of edges forming noncrossing chords. A book embedding is a circular permutation of the vertices together with a partition of the edges into pages. The pagenumber t(G) (also called book thickness) is the minimum number of pages in a book embedding of G. We present a general construction showing t(K_m,n) ? ?(m + 2n)/4?, which we conjecture optimal. We prove a result suggesting this is optimal for m ? 2n ? 3. For the most difficult case m = n, we consider vertex permutations that are regular, i.e., place vertices from each partite set into runs of equal size. Book embeddings with such orderings require ?(7n ? 2)/9? pages, which is achievable. The general construction uses fewer pages, but with an irregular ordering.     

Gray code enumeration of families of integer partitions

In this paper we show that the elements of certain families of integer partitions can be listed in a minimal change, or Gray code, order. In particular, we construct Gray code listings for the classes P_δ(n, k) and D(n, k) of partitions of n into parts of size at most k in which, for P_δ(n, k), the parts are congruent to one modulo δ and, for D(n, k), the parts are distinct. It is shown that the elements of these classes can be listed so that the only change between successive partitions is the increase of one part by δ (or the addition of δ ones) and the decrease of one part by δ (or the removal of δ ones), where, in the case of D(n, k), δ = 1.     

A class of solutions to the gossip problem, part III

We complete the study of NOHO-graphs, begun in Parts I and II of this paper. NOHO- graphs correspond to solutions to the gossip problem where No One Hears his Own information. These are graphs with a linear ordering on their edges such that an increasing path exists from each vertex to every other, but from no vertex to itself. We discard the two such graphs with no 2-valent vertices. In Part I, we translated these graphs into quadruples of integer sequences. In Part II, we characterized and enumerated the realizable quadruples and various subclasses of them. In Part III, we eliminate the overcounting of isomorphic graphs and obtain recurrence relations and generating functions to enumerate the non-isomorphic NOHO-graphs. If u_m=(1,1,2,…) satisfies u_m=3u_m-1–u_m-3, then the number of non-isomorphic NOHO- graphs on 2m+2 vertices is (u_m + u_[m/2]+1 + u_[m/2]+1 - u_[m/2]). We also examine some re lated questions.     

The quantitative interpretation of peaks in photoelectron spectra obtained with dispersive electrostatic analysers

General expressions are obtained which show the relation between the height and area of a photoelectron peak and the partial photoionisation cross-section. It is shown that, if the height is used as a measure of intensity, the energy spread of the incident photon beam and the analyser bandpass must be considered. The importance of having a complete knowledge of the variation of analyser efficiency with electron energy is stressed, the cylindrical mirror analyser being given as an example.     

Transition metal ion complexes of thiosemicarbazones derived from 2-acetylpyridineN-oxide. Part III. Spectral and thermal studies of the metal complexes of 2-acetylpyridineN-oxide thiosemicarbazone

Summary Metal complexes of 2-acetylpyridineN-oxide thiosemicarbazone have been prepared and characterized. Coordination occurs as a neutral bidentate (ON) and a deprotonated tridentate (ONS) ligand.     

Future contributions to Journal of Crystallographic and Spectroscopic Research

Future contributions to Journal of Crystallographic and Spectroscopic Research