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Even maps, the Colin de~Verdière number and representations of graphs

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 $σ$ is much more geometric/topological directly reflecting embeddability properties of the graph. They proved $μ(G) \leq σ(G) + 2$ and conjectured $μ(G) \leq σ(G)$ for any graph $G$. We confirm this conjecture. As far as we know, this is the first topological upper bound on $μ(G)$ 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 $μ(G) \leq 18$ and $σ(G)\geq 20$. We show that the gap appears on much smaller values, namely, we exhibit a graph $H$ for which $μ(H)\leq 7$ and $σ(H)\geq 8$. We also prove that, in general, the gap can be large: The incidence graphs $H_q$ of finite projective planes of order $q$ satisfy $μ(H_q) \in O(q^{3/2})$ and $σ(H_q) \geq q^2$.