Even maps, the Colin de Verdière number and representations of graphs Kaluza, Vojtech ; https://orcid.org/0000-0002-2512-8698 Tancer, Martin ; https://orcid.org/0000-0002-1191-6714 ddc:514 ddc:516 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 between μ and σ does not hold in general as van der Holst and Pendavingh showed that there is a graph G with μ(G) ≤ 18 and σ(G) ≥ 20. We show that the gap appears at much smaller values, namely, we exhibit a graph H for which μ(H) ≥ 7 and σ(H) ≥ 8. We also prove that, in general, the gap can be large: The incidence graphs Hq of finite projective planes of order q satisfy μ(Hq) ∈ O(q3/2) and σ(Hq) ≥ q2. Springer Nature 2022 info:eu-repo/semantics/article doc-type:Article Article http://purl.org/coar/resource_type/c_2df8fbb1 https://research-explorer.ista.ac.at/record/10335 Kaluza V, Tancer M. Even maps, the Colin de Verdière number and representations of graphs. <i>Combinatorica</i>. 2022;42:1317-1345. doi:<a href="https://doi.org/10.1007/s00493-021-4443-7">10.1007/s00493-021-4443-7</a> eng info:eu-repo/semantics/altIdentifier/doi/10.1007/s00493-021-4443-7 info:eu-repo/semantics/altIdentifier/issn/0209-9683 info:eu-repo/semantics/altIdentifier/wos/000798210100003 info:eu-repo/semantics/altIdentifier/arxiv/1907.05055 info:eu-repo/semantics/openAccess