Almost Selfcomplementary Graphs and Extensions

Abstract

<p>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.</p> <p>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.</p> <p>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.</p>