### Abstract

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 {±1,±2,...,±n} 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 {±1,±2,...,±n}. Using a constructive approach, we prove two combinatorial theorems of Tucker type. The theorems state 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 {0, 1}n +q for some integral vector q. These theorems are used to show in a constructive way the existence of an integral solution to a system of nonlinear equations under certain natural conditions. An economic application is provided.

Original language | English |
---|---|

Pages (from-to) | 391-407 |

Journal | Journal of Optimization Theory and Applications |

Volume | 144 |

Publication status | Published - 2010 |

### Fingerprint

### Cite this

*Journal of Optimization Theory and Applications*,

*144*, 391-407.

}

*Journal of Optimization Theory and Applications*, vol. 144, pp. 391-407.

**Combinatorial integer labeling theorems on finite sets with applications.** / van der Laan, G.; Talman, A.J.J.; Yang, Z.F.

Research output: Contribution to journal › Article › Scientific › peer-review

TY - JOUR

T1 - Combinatorial integer labeling theorems on finite sets with applications

AU - van der Laan, G.

AU - Talman, A.J.J.

AU - Yang, Z.F.

N1 - Appeared earlier as CentER DP 2007-88 (rt)

PY - 2010

Y1 - 2010

N2 - Tucker’s well-known combinatorial lemma states that, for any given symmetrictriangulation of the n-dimensional unit cube and for any integer labeling thatassigns to each vertex of the triangulation a label from the set {±1,±2,...,±n} 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 {±1,±2,...,±n}. Using a constructive approach, we prove two combinatorial theorems of Tucker type. The theorems state 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 {0, 1}n +q for some integral vector q. These theorems are used to show in a constructive way the existence of an integral solution to a system of nonlinear equations under certain natural conditions. An economic application is provided.

AB - Tucker’s well-known combinatorial lemma states that, for any given symmetrictriangulation of the n-dimensional unit cube and for any integer labeling thatassigns to each vertex of the triangulation a label from the set {±1,±2,...,±n} 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 {±1,±2,...,±n}. Using a constructive approach, we prove two combinatorial theorems of Tucker type. The theorems state 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 {0, 1}n +q for some integral vector q. These theorems are used to show in a constructive way the existence of an integral solution to a system of nonlinear equations under certain natural conditions. An economic application is provided.

M3 - Article

VL - 144

SP - 391

EP - 407

JO - Journal of Optimization Theory and Applications

JF - Journal of Optimization Theory and Applications

SN - 0022-3239

ER -