Crossing numbers of sequences of graphs II: Planar tiles

We describe a method of creating an infinite family of crossing-critical graphs from a single small planar map, the tile, by gluing together many copies of the tile together in a circular fashion. This method yields all known infinite families of k-crossing-critical graphs. Furthermore, the method yields new infinite families, which extend from (4,6) to (3.5,6) the interval of rationals r for which there is, for some k, an infinite sequence of k-crossing-critical graphs all having average degree r. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 332–341, 2003