Partitions of Complete Geometric Graphs into Plane Trees

Abstract

AbstractConsider the following open problem: 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 \(n-\sqrt{n/12}\) plane trees.KeywordsConvex HullSpan TreeEdge IncidentBoundary EdgeGeometric GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.