### 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 |
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Place of Publication | Tilburg |

Publisher | Operations research |

Number of pages | 21 |

Volume | 2007-88 |

Publication status | Published - 2007 |

### Publication series

Name | CentER Discussion Paper |
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Volume | 2007-88 |

### Keywords

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

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## 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.