Combinatorial integer labeling theorems on finite sets with applications

G. van der Laan, A.J.J. Talman, Z. 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 {±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}
    Original languageEnglish
    Pages (from-to)391-407
    JournalJournal of Optimization Theory and Applications
    Volume144
    DOIs
    Publication statusPublished - 2010

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