The Colin de Verdiere parameter ?(G) is known to capture some of the

embeddability properties of a graph G: For instance, ?(G) ? 3 if and only if

G is planar and ?(G) ? 4 if and only if G is linkless embeddable. In order

to study topological properties of graphs G with ?(G) > 4, van der Holst and

Pendavingh introduced a parameter ?(G). They showed that ?(G) ? k if and

only if ?(G) ? k for k = 1, 2, 3, 4. In addition, they proved that ?(G) ? ?(G)+2

for every G and conjectured that ?(G) ? ?(G) in general. They also showed

that the two parameters may differ.

In this talk, after an introduction on the parameters ? and ?, I plan to

present a proof of the aforementioned conjecture of van der Holst and Pen-

davingh. This is a joint work with Martin Tancer.

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