Even Maps, the Colin de Verdière Number and Representations of Graphs

@article{Kaluza2020EvenMT,
  title={Even Maps, the Colin de Verdi{\`e}re Number and Representations of Graphs},
  author={Vojtech Kaluza and Martin Tancer},
  journal={Combinatorica},
  year={2020},
  pages={1-29}
}
Van der Holst and Pendavingh introduced a graph parameter σ , which coincides with the more famous Colin de Verdière graph parameter μ for small values. However, the definition of a is much more geometric/topological directly reflecting embeddability properties of the graph. They proved μ ( G ) ≤ σ ( G ) + 2 and conjectured σ ( G ) ≤ σ ( G ) for any graph G . We confirm this conjecture. As far as we know, this is the first topological upper bound on σ ( G ) which is, in general, tight. Equality… 

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