  The game of Nim as played on graphs was introduced in Nim on Graphs I and
extended in Nim on Graphs II by Masahiko Fukuyama. His papers detail the
calculation of Grundy numbers for graphs under specific circumstances. We
extend these results and introduce the strategy for even cycles. This paper
examines a more general class of graphs by restricting the edge weight to one.
We provide structural conditions for which there exist a winning strategy. This
yields the solution for the complete graph.
