Partitions of Complete Geometric Graphs into Plane Trees

Consider the following question: does every complete geometric graph K2n have a partition of its edge set into n plane spanning trees? We approach this problem from three directions. First, we study the case of convex geometric graphs. It is well known that the complete convex graph K2n has a partition into n plane spanning trees. We characterise all such partitions. Second, we give a sufficient condition, which generalises the convex case, for a complete geometric graph to have a partition into plane spanning trees. Finally, we consider a relaxation of the problem in which the trees of the partition are not necessarily spanning. We prove that every complete geometric graph Kn can be partitioned into at most n − n/12 plane trees. This is the best known bound even for partitions into plane subgraphs. Key words: geometric graph, complete graph, plane tree, convex graph, book embedding, book thickness, crossing family Preprint submitted to Computational Geometry 9 August 2005 

• Type: Conference
• Year: 2004
• Where: GD
• Authors: Prosenjit Bose, Ferran Hurtado, Eduardo Rivera-Campo, David R. Wood
• Keywords: Complete Geometric Graph, GD 2004, Geometric Graph, Plane Spanning Trees