Partitions of complete geometric graphs into plane trees
Consider the following question: does every complete geometric graph K 2n 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 K 2n 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 K n can be partitioned into at most n-√n/12 plane trees. This is the best known bound even for partitions into plane subgraphs.
|Keywords||Book embedding, Book thickness, Complete graph, Convex graph, Crossing family, Geometric graph, Plane tree|
Bose, P, Hurtado, F. (Ferran), Rivera-Campo, E. (Eduardo), & Wood, D. (2006). Partitions of complete geometric graphs into plane trees. Computational Geometry, 34(2), 116–125. doi:10.1016/j.comgeo.2005.08.006