New Infinite Families of Almost-Planar Crossing-Critical Graphs

We show that, for all choices of integers k > 2 and m, there are simple 3connected 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 at least 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 ) .