Nim-Regularity of Graphs

Abstract

Ehrenborg and Steingrímsson defined simplicial Nim, and defined Nim-regular complexes to be simplicial complexes for which simplicial Nim has a particular type of winning strategy. We completely characterize the Nim-regular graphs by the exclusion of two vertex-induced subgraphs, the graph on three vertices with one edge and the graph on five vertices which is complete except for one missing edge. We show that all Nim-regular graphs have as their basis the set of disjoint unions of circuits (minimal non-faces) of the graph.