# Combinatorial Integer Labeling Thorems on Finite Sets with an Application to Discrete Systems of Nonlinear Equations

G. van der Laan, A.J.J. Talman, Z.F. Yang

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### Abstract

Tucker's well-known combinatorial lemma states that for any given symmetric triangulation of the n-dimensional unit cube and for any integer labeling that assigns to each vertex of the triangulation a label from the set f§1;§2; ¢ ¢ ¢ ;§ng with the property that antipodal vertices on the boundary of the cube are assigned opposite labels, the triangulation admits a 1-dimensional simplex whose two vertices have opposite labels. In this paper we are concerned with an arbitrary finite set D of integral vectors in the n-dimensional Euclidean space and an integer labeling that assigns to each element of D a label from the set f§1;§2; ¢ ¢ ¢ ;§ng. Using a constructive approach we prove two combinatorial theorems of Tucker type, stating that under some mild conditions there exists two integral vectors in D having opposite labels and being cell-connected in the sense that both belong to the set f0; 1gn+q for some integral vector q. These theorems will be used to show in a constructive way the existence of an integral solution to a system of nonlinear equations under certain natural conditions.
Original language English Tilburg Operations research 21 2007-88 Published - 2007

### Publication series

Name CentER Discussion Paper 2007-88

### Keywords

• Sperner lemma
• Tucker lemma
• integer labeling
• simplicial algorithm
• discrete nonlinear equations

• ## Cite this

van der Laan, G., Talman, A. J. J., & Yang, Z. F. (2007). Combinatorial Integer Labeling Thorems on Finite Sets with an Application to Discrete Systems of Nonlinear Equations. (CentER Discussion Paper; Vol. 2007-88). Operations research.