A complete multipartite graph is one in which there exists a
partition of the vertices so that two vertices are joined by
an edge if and only if they occur in different parts of the
partition. A hamilton decomposition of a graph G is a partition
of the edges of G, each element of which induces a hamilton cycle.
It is known that complete multipartite graphs have a hamilton
decomposition if and only if each of the parts has the same number
of vertices and each vertex has even degree.

In this talk, hamilton decompositions are found which are "fair"
in the sense that between each pair of parts of vertices, the
edges are shared as evenly as possible among the hamilton cycles.
The method of proof uses amalgamations (graph homomorphisms), a
method which is of interest in its own right.
On an unrelated topic, if there is time, an open problem in the
area of "Group Testing" will be described. The aim is to minimize
the largest number of tests that may be required to identify
contaminated samples among a family of samples. Enquiries to Catherine Greenhill, 9385 7105, csg@unsw.edu.au