The Metamorphosis of 2-fold Triple Systems into Maximum Packings of 2Kn with 4-cycles

Abstract

The graph is called a hinge. A hinge system of order n is a pair (X, H) where H is a collection of edge disjoint hinges which partition the edge set of 2Kn with vertex set X. Let (X, H) be a hinge system and D the collection of double edges from the hinges. Let H*= (= the 4-cycles left over when the double edges are removed). If the edges of D can be arranged into a collection of 4-cycles D*, then (X, H* D*) is a 2-fold 4-cycle system called a metamorphosis of (X, H) into (X, H* union C*). In a previous work, it was shown that the spectrum for hinge systems having a metamorphosis into a 2-fold 4-cycle system is precisely the set of all n congruent to 0, 1, 4, or 9 (mod 12). In this thesis, we extend that result by showing that the spectrum for hinge systems having a metamorphosis into a maximum packing of 2Kn with 4-cycles is precisely the set of all n congruent to 3, 6, 7 or 10 (mod 12). No such systems exist for n = 6 or 7. We point out that if we partition each hinge in a hinge system into a pair of triangles, we have a 2-fold triple system, hence, the title of this thesis.