A borsuk theorem for antipodal links and a spectral characterization of linklessly embeddable graphs

László Lovász, Alexander Schrijver

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37 Citations (Scopus)

Abstract

For any undirected graph G, let μ(G) be the graph parameter introduced by Colin de Verdière. In this paper we show that μ(G) ≤ 4 if and only if G is linklessly embeddable (in R3). This forms a spectral characterization of linklessly embeddable graphs, and was conjectured by Robertson, Seymour, and Thomas. A key ingredient is a Borsuk-type theorem on the existence of a pair of antipodal linked (k-1)-spheres in certain mappings φ : S2k → R2k-1. This result might be of interest in its own right. We also derive that λ(G) ≤ 4 for each linklessly embeddable graph G = (V, E), where A(G) is the graph parameter introduced by van der Holst, Laurent, and Schrijver. (It is the largest dimension of any subspace L of RV such that for each nonzero x ε L, the positive support of x induces a nonempty connected subgraph of G.)

Original languageEnglish
Pages (from-to)1275-1285
Number of pages11
JournalProceedings of the American Mathematical Society
Volume126
Issue number5
Publication statusPublished - Dec 1 1998

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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