Algorithms to Find Completely-Labelled Simplices

Abstract

While the proof of Sperner’s lemma given in 1.4 does not suggest any method to find completely-labelled simplices, we noted in example I.4.3 that n-simplices of G with the labels 0,1,...,n-1 formed paths. Cohen [6] gave a proof of Sperner’s lemma based on these paths; we present his argument in Section 1. Cohen’s proof is inductive—we still have the problem of how to start. Two possible methods will be given in Section 2. Sections 3 and 4 show how these methods can be implemented using K₂ (m). Section 5 describes Scarf’s algorithm, and Section 6 illustrates the algorithms with an example.