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 straightline 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}. 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. Communicated by G. Liotta and S. H. Whitesides; submitted November 1998; ...

• Type: Conference
• Year: 1998
• Where: GD
• Authors: Michael B. Dillencourt, David Eppstein, Daniel S. Hirschberg
• Keywords: Complete Bipartite Graphs, Computer Graphics, GD 1998, Geometric Thickness, Geometric Thickness Coincides