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Combinatorial integer labeling theorems on finite sets with applications
van der Laan,G. Talman,A.J.J. Yang,Z.F.
van der Laan,G.
Talman,A.J.J.
Yang,Z.F.
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 {±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.
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Appeared earlier as CentER DP 2007-88 (rt)
Date
2010
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JOTA144.pdf
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van der Laan, G, Talman, A J J & Yang, Z F 2010, 'Combinatorial integer labeling theorems on finite sets with applications', Journal of Optimization Theory and Applications, vol. 144, pp. 391-407.
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info:eu-repo/semantics/restrictedAccess