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

For any undirected graph G, let mu(G) be the graph parameter introduced by Colin de Verdiere. In this paper we show that mu(G) less than or equal to 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 phi : S-2k --> R2k-1. This result might be of interest in its own right. We also derive that lambda(G) less than or equal to 4 for each linklessly embeddable graph G = (V, E), where lambda(G) is the graph parameter introduced by van der Hoist, Laurent, and Schrijver. (It is the largest dimension of any subspace L of R-V such that for each nonzero x is an element of L, the positive support of x induces a nonempty connected subgraph of G.).