Discussiones Mathematicae Graph Theory 26(3) (2006) 439-338
doi: 10.7151/dmgt.1336

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COMBINATORIAL LEMMAS FOR POLYHEDRONS I

Adam Idzik

Akademia Świetokrzyska
Świetokrzyska 15, 25-406 Kielce, Poland
and
Institute of Computer Science, Polish Academy of Sciences
Ordona 21, 01-237 Warsaw, Poland
e-mail: adidzik@ipipan.waw.pl

Konstanty Junosza-Szaniawski

Warsaw University of Technology
Pl. Politechniki 1, 00-661 Warsaw, Poland
e-mail: k.szaniawski@mini.pw.edu.pl

Abstract

We formulate general boundary conditions for a labelling of vertices of a triangulation of a polyhedron by vectors to assure the existence of a balanced simplex. The condition is not for each vertex separately, but for a set of vertices of each boundary simplex. This allows us to formulate a theorem, which is more general than the Sperner lemma and theorems of Shapley; Idzik and Junosza-Szaniawski; van der Laan, Talman and Yang. A generalization of the Poincaré-Miranda theorem is also derived.

Keywords: b-balanced simplex, labelling, polyhedron, simplicial complex, Sperner lemma.

2000 Mathematics Subject Classification: 05B30, 47H10, 52A20, 54H25.

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Received 2 December 2005
Revised 3 October 2006