Sperner labellings: A combinatorial approach

In 2002, De Loera, Peterson and Su proved the following conjecture of Atanassov: let T be a triangulation of a d-dimensional polytope P with n vertices v1, v2, . . . , vn; label the vertices of T by 1, 2, ..., n in such a way that a vertex of T belonging to the interior of a face F of P can only be labelled by j if vj is on F; then there are at least n - d simplices labelled with d + 1 dierent labels. We prove a generalization of this theorem which renes this lower bound and which is valid for a larger class of objects. Key Words: chain map; fully-labelled simplex; labelling; polytopal body; polytope; Sperner's lemma; triangulation.

• Type: Journal
• Year: 2006
• Where: JCT
• Authors: Frédéric Meunier
• Keywords: D-dimensional Polytope, De Loera, JCT 2006, Labels