# Surgical Techniques for Constructing Minimal Orientable Imbeddings of Joins and Compositions of Graphs

6-1991

## Degree Name

Doctor of Philosophy

## Department

Mathematics

Dr. Arthur T. White

Dr. Allen Schwenk

Dr. Erik Schreiner

Dr. Ajay Gupta

## Abstract

The various cases within the proof of the Heawood Map-Coloring Theorem, which established the genus of the complete graphs, utilized various techniques--some for the first time. This activity spurred interest in determining the genus of various other classes of graphs. However, very few generally applicable techniques have been developed, beyond those used in the proof of this famous theorem. Finding genera of arbitrary graphs remains a very difficult problem.

In this dissertation, we describe two surgical techniques for imbedding graphs. The first construction, called a graphical surface,views an orientable surface as a fattened graph, i.e., vertices become spheres and edges become tubes. We describe how a graph is imbedded in such a surface using a combinatorial structure which we call an arcchain (an ordered set of oriented objects). The second construction, called a crown, is primarily used to add clones of individual vertices of an imbedded graph. These constructions are developed independently but can be used in conjunction with one another.

Our results on joins include new (surgical) proofs of the well known genus formulae for complete bipartite graphs and regular complete 4-partite graphs. Among the new results are several genus formulae for graphs of the form $\bar{\rm K}\sb{\rm a}$ + G including large classes of complete tripartite graphs. We also establish genus formulae for all joins of cycles and paths.

A brief study of compositions completes our work. For compositions G(H) we require H to have even order. It is shown that under modest restrictions on G, the genus of G(H) is independent of the size and structure of H. Other results restrict H but require only that G be connected and triangle-free. We also define the concept of a generalized composition graph and determine genera for several classes of these.