Thickness and colorability of geometric graphs

The geometric thickness of a graph G is the smallest integer t such that there exist a straight-line drawing Γ of G and a partition of its straight-line edges into t subsets, where each subset induces a planar drawing in Γ. Over a decade ago, Hutchinson, Shermer, and Vince proved that any n  -vertex graph with geometric thickness two can have at most 6n−186n−18 edges, and for every n≥8n≥8 they constructed a geometric thickness-two graph with 6n−206n−20 edges. In this paper, we construct geometric thickness-two graphs with 6n−196n−19 edges for every n≥9n≥9, which improves the previously known 6n−206n−20 lower bound. We then construct a thickness-two graph with 10 vertices that has geometric thickness three, and prove that the problem of recognizing geometric thickness-two graphs is NP-hard, answering two questions posed by Dillencourt, Eppstein and Hirschberg. Finally, we prove the NP-hardness of coloring graphs of geometric thickness t   with 4t−14t−1 colors, which strengthens a result of McGrae and Zito, when t=2t=2.