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

Abstract

For any undirected graph G,let µ(G) be the graph parameter in- troduced by Colin de Verdi`ere. In this paper we show that µ(G) ≤ 4if and only if G is linklessly embeddable (in R3). This forms a spectral character- ization 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 λ(G) is the graph parameter introduced by van der Holst, Lau- rent, and Schrijver. (It is the largest dimension of any subspace L of RV such that for each nonzero x ∈ L, the positive support of x induces a nonempty connected subgraph of G.)