
NAME:
Kira Owsley
EvenCycle 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 nonprimitive
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 6cycle systems, if Kn has a primitive 6cycle
system, then Kn+12 also has a primitive 6cycle system, and make
other progress in proving if Kn has a primitive mcycle system form
even, then Kn+2m has a primitivemcycle system.



