On the Sperner lemma and some colorings of graphs

In this journal, Daniel I. A. Cohen 2] gave a proof of the strong Sperner lemma based on “search” techniques of paths. Klaus Wagner 5] associates a particular tree to a labeled subdivision of a geometric simplex and proves that lemma by counting the number of vertices with an odd degree of that tree. In this paper we prove a general theorem about colorings of simplicial complexes by using a method similar to Wagner's. This theorem implies the strong Sperner lemma and can be used to study some colorings of graphs.