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NAME: Kira Owsley

Even-Cycle Decompositions with no Subsystem

The decomposition of edges of a graph is a traditional area of research in graph theory.For a graph G with vertex set V and edge set E , a decomposition of G is a partition of E .Assume that E contains all possible edges connecting vertices in V , and let C be a decompositionofG . We say thatC is non-primitive if there exists a nontrivial proper subsetof vertices V 0 V and a proper subset of parts C0 C such that C0 is itself a decompositionof all edges connecting vertices in V 0. In this paper, we investigate the existence ofprimitive cycle systems – primitive decompositions into cycles of a fixed length – wherethe cycle size is even. We show that K9 – a graph on 9 vertices with all possible edges – and K21 have primitive 6-cycle systems, if Kn has a primitive 6-cycle system, then Kn+12 also has a primitive 6-cycle system, and make other progress in proving if Kn has a primitive m-cycle system form even, then Kn+2m has a primitivem-cycle system.