Some Properties of a Generalized Sperner Labeling

A generalization of Sperner’s labeling for simplices is considered. It allows us to give any label not only to points from the interior of the simplex but also to points from the relative interior of each facet, while the Sperner labeling rule is preserved for all points on the boundary of each facet. Some properties of this labeling and its behavior on the facets of the simplex are discussed. Also, necessary and sufficient conditions for the existence of an odd number of completely labelled simplices in any triangulation of the simplex are given.