@techreport{264c28a510b644e196944b2c4b7c5c7a,

title = "Combinatorial Integer Labeling Thorems on Finite Sets with an Application to Discrete Systems of Nonlinear Equations",

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.",

keywords = "Sperner lemma, Tucker lemma, integer labeling, simplicial algorithm, discrete nonlinear equations",

author = "{van der Laan}, G. and A.J.J. Talman and Z.F. Yang",

note = "Subsequently published in Journal of Optimization Theory and Applications, 2010 (rt) Pagination: 21",

year = "2007",

language = "English",

volume = "2007-88",

series = "CentER Discussion Paper",

publisher = "Operations research",

type = "WorkingPaper",

institution = "Operations research",

}