New infinite families of almost-planar crossing-critical graphs

We show that, for all choices of integers k > 2 and m, there are simple 3-connected k-crossing-critical graphs containing more than m vertices of each even degree <= 2k - 2. This construction answers one half of a question raised by Bokal, while the other half asking analogously about vertices of odd degrees atleast 7 in crossing-critical graphs remains open. Furthermore, our newly constructed graphs have several other interesting properties; for instance, they are almost planar and their average degree can attain any rational value in the interval [3 + 1/5, 6 - 8/k+1).