TITLE: Do there exist sparse graphs with large crossing number?
QUESTION [13 upvotes]: Does there exist a sequence of graphs $\{ G_n \}$ such that

$G_n$ has $n$ vertices,
the number of edges of $G_n$ is $O(n)$, and
the crossing number of $G_n$ is $\Omega(n)$?

In particular, do random $k$-regular graphs satisfy this?
Motivation: the crossing number inequality gives a lower bound on the crossing number $cr(G)$, just in terms of the number of vertices $v$ and edges $e$. In particular, if $e \ge 4v$ then
$$ cr(G) \ge \frac{e^3}{64v^2}.$$
This is tight, up to a constant factor, meaning there are graphs whose crossing numbers are approximately this small. But this still raises the question of how large the crossing number can be, particularly for sparse graphs.

REPLY [5 votes]: Actually, even the crossing number of the random 3-regular graph on $n$ vertices has crossing number $\Theta(n^2)$. This follows from the lower bound obtained by Bollobás for the isoperimetric number of random regular graphs (European Journal of Combinatorics, Vol. 9 (1988), pp. 241-244); the random $k$-regular graph (for any fixed $k$) thus has large bisection width, guaranteeing a large (i.e., $\Theta(n^2)$) crossing number.