Nim On The Complete Graph

The two-player game of Nim on graphs is played on a regular graph with positively weighted edges by moving alternately from a fixed starting vertex to an adjacent vertex, decreasing the weight of the incident edge to a strictly smaller non-negative integer. The game ends when a player is unable to move since all edges incident with the vertex from which the player is to move have weight zero. In this paper we extend Fukuyama’s results to include a strategy for even cycles, and we give a solution to the complete graph when each edge has weight one. We also consider the recent strides made in the complete graph with arbitrary weight.