Thickness and Colorability of Geometric Graphs

The geometric thickness θ(G) 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 − 18 edges, and for every n ≥ 8 they constructed a geometric thickness two graph with 6n − 20 edges. In this paper, we construct geometric thickness two graphs with 6n − 19 edges for every n ≥ 9, which improves the previously known 6n − 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 − 1 colors, which strengthens a result of McGrae and Zito, when t = 2.