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) ≤ σ(G) + 2 and conjectured µ(G) ≤ 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) ≤ 18 and σ(G) ≥ 20. We show that the gap appears on 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 H q of finite projective planes of order q satisfy µ(H q ) ∈ O(q 3/2 ) and σ(H q ) ≥ q 2 .