Margaret Morton

Embedding Digraphs on Orientable Surfaces

Abstract We consider a notion of embedding digraphs on orientable surfaces, applicable to digraphs in which the indegree equals the outdegree for every vertex, i.e., Eulerian digraphs. This idea has been considered before in the context of compatible Euler tours or orthogonal A-trails by Andersen and by Bouchet. This prior work has mostly been limited to embeddings of Eulerian digraphs on predetermined surfaces and to digraphs with underlying graphs of maximum degree at most 4. In this paper, a foundation is laid for the study of all Eulerian digraph embeddings. Results are proved which are analogous to those fundamental to the theory of undirected graph embeddings, such as Dukes theorem [5], and an infinite family of digraphs which demonstrates that the genus range for an embeddable digraph can be any nonnegative integer given. We show that it is possible to have genus range equal to one, with arbitrarily large minimum genus, unlike in the undirected case. The difference between the minimum genera of a digraph and its underlying graph is considered, as is the difference between the maximum genera. We say that a digraph is upper-embeddable if it can be embedded with two or three regions and prove that every regular tournament is upper-embeddable.

Classification of 4- and 5-arc-transitive cubic graphs of small girth

This paper classifies all finite connected 4- and 5-arc-transitive cubic graphs that contain circuits of length less than or equal to 11, or of length 13, and some of those graphs with circuits of length 12

A Comparative Study of Two Nationwide Examinations: Maths with Calculus and Maths with Statistics.

In recent years a group at Auckland University has made an analysis of the results in the nationwide Year 12 calculus examination and the ‘Equity in Mathematics Education’ team in Wellington has done a similar investigation of the corresponding statistics paper. Both groups initially analysed the raw (unscaled) marks from the candidates for the 1987 and 1988 examinations on the basis of gender, school authority and school type. Of these three effects the first two were significant for the statistics examination and the last two for the calculus paper. The candidates were then divided into cohorts depending on whether they took both or just one of these mathematics papers. The effect of the number of mathematics papers a student took was highly significant, consequently the three-way analysis was repeated for each cohort. Gender was then no longer significant in either study but type and authority were significant for both analyses. An overview of the effect of gender in a question by question analysis of each examination has been included.

Generalized Steinhaus graphs

A generalized Steinhaus graph of order n and type s is a graph with n vertices whose adjacency matrix (ai,j) satisfies the relation where 2 ≦i≦n−1, i + s(i − 1 ≦ j ≦ n, cr,i,j ϵ {0,1} for all 0 ≦ r ≦ s(i) −1 and cs(i)−1,i,j = 1. The values of s(i) and cr,i,j are fixed but arbitrary. Generalized Steinhaus graphs in which each edge has probability ½ are considered. In an article by A. Blass and F. Harary [“Properties of Almost All Graphs and Complexes,” Journal of Graph Theory, Vol. 3 (1976), pp. 225–240], a collection of first-order axioms are given from which any first-order property in graph theory or its negation can be deduced. We show that almost all generalized Steinhaus graphs satisfy these axioms. Thus the first-order theory of random generalized Steinhaus graphs is identical with the theory of random graphs. Quasi-random properties of graphs are described by F. R. K. Chung, R. L. Graham, and R. M. Wilson, [“Quasi-Random Graphs,” Combinatorica, Vol. 9 (1989), pp. 345–362]. We conclude by demonstrating that almost all generalized Steinhaus graphs obey Property 2 and hence all equivalent quasi-random properties.