Stars and bonds in crossing-critical graphs

The structure of all known infinite families of crossing–critical graphs has led to the conjecture that crossing–critical graphs have bounded bandwidth. If true, this would imply that crossing–critical graphs have bounded degree, that is, that they cannot contain subdivisions of K1,n for arbitrarily large n. In this paper we prove two results that revolve around this conjecture. On the positive side, we show that crossing–critical graphs cannot contain subdivisions of K2,n for arbitrarily large n. On the negative side, we show that there are graphs with arbitrarily large maximum degree that are 2-crossing–critical in the projective plane