### 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 R^{3}). 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 : S^{2k} → R^{2k-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 R^{V} such that for each nonzero x ε L, the positive support of x induces a nonempty connected subgraph of G.)

Original language | English |
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Pages (from-to) | 1275-1285 |

Number of pages | 11 |

Journal | Proceedings of the American Mathematical Society |

Volume | 126 |

Issue number | 5 |

Publication status | Published - Dec 1 1998 |

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Proceedings of the American Mathematical Society*,

*126*(5), 1275-1285.