Abstract
Consider 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.
Research of all the authors was completed in the Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Barcelona, Spain. Research of P. Bose supported by NSERC. Research of F. Hurtado supported by projects DURSI 2001SGR00224, MCYT-BFM2001-2340, MCYT-BFM2003-0368 and Gen. Cat 2001SGR00224. Research of E. Rivera-Campo supported by MECD, Spain and Conacyt, México. Research of D. R. Wood supported by NSERC and COMBSTRU.
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© 2005 Springer-Verlag Berlin Heidelberg
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Bose, P., Hurtado, F., Rivera-Campo, E., Wood, D.R. (2005). Partitions of Complete Geometric Graphs into Plane Trees. In: Pach, J. (eds) Graph Drawing. GD 2004. Lecture Notes in Computer Science, vol 3383. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31843-9_9
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DOI: https://doi.org/10.1007/978-3-540-31843-9_9
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