Geometric Thickness of Complete Graphs
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 ⌈(n/5.646) +0.342⌉ and ⌈n/4⌉, and we give exact values of the geometric thickness of K n for n ≤ 12 and n ∈ 15,16.
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