Date of Award

5-1989

Degree Type

Thesis

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Supervisor

Dr. Alexander Rosa

Abstract

In this thesis the concept of selfcomplementary graphs is extended to almost selfcomplementary graphs. We dine a p-vertex graph to be almost selfcomplementary if it is isomorphic to its complement with respect to Kp-e, the complete graph with one edge deleted. An almost selfcomplementary graph with p vertices exists if and only if p=2 or 3 (mod 4), ie., precisely when selfcomplementary graphs do not exist. We investigate various properties of almost selfcomplementary graphs and examine the similarities and differences with those of selfcomplementary graphs.

The concepts of selfcomplementary and almost selfcomplementary graphs are combined to define so-called k-selfcomplementary graphs which include the former two classes as subclasses. Although a k-selfcomplementary graph may contain fewer edges than a selfcomplementary or an almost selfcomplementary graph it is found that the former preserves most of the properties of the latter graphs.

The notion of selfcomplementarity is further extended to combinatorial designs. In particular, we examine whether a Steiner triple system (twofold triple system, and a Steiner system S(2,4,v), respectively) can be partitioned into two isomorphic hypergraphs.


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