  We define the geometric thickness of a graph to be the smallest number of
layers such that we can draw the graph in the plane with straight-line edges
and assign each edge to a layer so that no two edges on the same layer cross.
The geometric thickness lies between two previously studied quantities, the
(graph-theoretical) thickness and the book thickness. We investigate the
geometric thickness of the family of complete graphs, K_n. We show that the
geometric thickness of K_n lies between ceiling((n/5.646) + 0.342) and
ceiling(n/4), and we give exact values of the geometric thickness of K_n for n
<= 12 and n in {15,16}. We also consider the geometric thickness of the family
of complete bipartite graphs. In particular, we show that, unlike the case of
complete graphs, there are complete bipartite graphs with arbitrarily large
numbers of vertices for which the geometric thickness coincides with the
standard graph-theoretical thickness.
