### Bounded-Degree Graphs have Arbitrarily Large Geometric Thickness

#### Abstract

The *geometric thickness* of a graph $G$ is the minimum integer $k$ such that there is a straight line drawing of $G$ with its edge set partitioned into $k$ plane subgraphs. Eppstein [Separating thickness from geometric thickness. In *Towards a Theory of Geometric Graphs*, vol. 342 of *Contemp. Math.*, AMS, 2004] asked whether every graph of bounded maximum degree has bounded geometric thickness. We answer this question in the negative, by proving that there exists $\Delta$-regular graphs with arbitrarily large geometric thickness. In particular, for all $\Delta\geq9$ and for all large $n$, there exists a $\Delta$-regular graph with geometric thickness at least $c\sqrt{\Delta}\,n^{1/2-4/\Delta-\epsilon}$. Analogous results concerning graph drawings with few edge slopes are also presented, thus solving open problems by Dujmović et al. [Really straight graph drawings. In *Proc. 12th International Symp. on Graph Drawing* (GD '04), vol. 3383 of *Lecture Notes in Comput. Sci.*, Springer, 2004] and Ambrus et al. [The slope parameter of graphs. Tech. Rep. MAT-2005-07, Department of Mathematics, Technical University of Denmark, 2005].