Sequences Characterizing k-Trees

  • Zvi Lotker
  • Debapriyo Majumdar
  • N. S. Narayanaswamy
  • Ingmar Weber
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4112)


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.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Zvi Lotker
    • 1
  • Debapriyo Majumdar
    • 2
  • N. S. Narayanaswamy
    • 3
  • Ingmar Weber
    • 2
  1. 1.Centrum voor Wiskunde en InformaticaAmsterdam
  2. 2.Max-Planck-Institut für InformatikSaarbrücken
  3. 3.Indian Institute of Technology MadrasChennai

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