Enclosings of Small Cycle Systems

Abstract

In 2003 Hurd and others considered the problem of enclosing a triple system TS(v; lambda) in a triple system TS(v + s; lambda + m), focusing on smallest possible enclosings. In the second chapter, their result is generalized using a new proof based on a graph-theoretic technique. Four constructions are presented; they are exhaustive in the sense that, for each possible congruence of the parameters v or s and m, at least one construction can be applied to obtain an enclosing. In each construction, the value of v or s is restricted. This naturally led to the question of whether or not a lambda-fold 4-cycle system could be enclosed for all possible values. In the third chapter, we completely solve the enclosing problem by construction for lambda-fold 4-cycle systems for u > 1.