Geometric thickness of complete graphs

Abstract

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, {Kn}. We show that the geometric thickness of Kn lies between ⌈(n=5.646) + 0.342⌉ and ⌈n=4⌉, and we give exact values of the geometric thickness of Kn for n ≤ 12 and n ∈ {15, 16}.