Thickness-two graphs part one: New nine-critical graphs, permuted layer graphs, and Catlin's graphs

The purpose of this paper is to offer new insight and tools toward the pursuit of the largest chromatic number in the class of thickness-two graphs. At present, the highest chromatic number known for a thickness-two graph is 9, and there is only one known color-critical 1 such graph. We introduce 40 small 9-critical thickness-two graphs, and then use a new construction, the permuted layer graphs, together with a construction of Hajós to create an infinite family of 9-critical thickness-two graphs. Finally, a non-trivial infinite subfamily of Catlin's graphs, with directly computable chromatic numbers, is shown to have thickness two.