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Sequences Characterizing k-Trees

MPG-Autoren
http://pubman.mpdl.mpg.de/cone/persons/resource/persons44944

Lotker,  Zvi
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

http://pubman.mpdl.mpg.de/cone/persons/resource/persons44972

Majumdar,  Debapriyo
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

http://pubman.mpdl.mpg.de/cone/persons/resource/persons45715

Weber,  Ingmar
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

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Zitation

Lotker, Z., Majumdar, D., Narayanaswamy, N., & Weber, I. (2006). Sequences Characterizing k-Trees. In Computing and Combinatorics, 12th Annual International Conference, COCOON 2006 (pp. 216-225). Berlin, Germany: Springer.


Zitierlink: http://hdl.handle.net/11858/00-001M-0000-000F-23EB-4
Zusammenfassung
A non-decreasing sequence of n integers is the degree sequence of a 1-tree (i.e., an ordinary tree) on n vertices if and only if there are least two 1’s in the sequence, and the sum of the elements is 2(n–1). We generalize this result in the following ways. First, a natural generalization of this statement is a necessary condition for k-trees, and we show that it is not sufficient for any k > 1. Second, we identify non-trivial sufficient conditions for the degree sequences of 2-trees. We also show that these sufficient conditions are almost necessary using bounds on the partition function p(n) and probabilistic methods. Third, we generalize the characterization of degrees of 1-trees in an elegant and counter-intuitive way to yield integer sequences that characterize k-trees, for all k.