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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)

Abstract

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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References

  1. 1.
    Duke, R.A., Winkler, P.M.: Degree Sets of k-Trees: Small k. Israel Journal of Mathematics 40(3-4) (1981)Google Scholar
  2. 2.
    Duke, R.A., Winkler, P.M.: Realizability of almost all degree sets by k-trees. Congressus Numerantium 35, 261–273 (1982)MathSciNetGoogle Scholar
  3. 3.
    Erdös, P., Gallai, T.: Graphs with prescribed degree of vertices (Hungarian). Mat. Lapok 11, 264–274 (1960)Google Scholar
  4. 4.
    Erdös, P., Simonovits, M.: Compactness results in extremal graph theory. Combinatorica 2, 275–288 (1982)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Freuder, E.C.: Complexity of k-tree structured constraint satisfaction problems. In: Proc. of the 8th National Conference on Artificial Intelligence (1990)Google Scholar
  6. 6.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, London (1980)MATHGoogle Scholar
  7. 7.
    Hakimi, S.L.: On the realizability of a set of integers as degrees of the vertices of a graph. J. SIAM Appl. Math. 10, 496–506 (1962)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Hammer, P.L., Simeone, B.: The splittance of a graph. Combinatorica 1, 275–284 (1981)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Havel, V.: A remark on the existence of finite graphs (Czech). Casopis Pest. Mat. 80, 477–480 (1955)MATHMathSciNetGoogle Scholar
  10. 10.
    Hardy, G.H., Ramanujan, S.: Une formule asymptotique pour le nombres des partitions de n. Comptes Rendus Acad. Sci. Paris, Ser. A (January 2, 1917)Google Scholar
  11. 11.
    Hardy, G.H., Ramanujan, S.: Asymptotic formulae in combinatory analysis. Proc. London Math. Soc. 17, 75–115 (1918)CrossRefGoogle Scholar
  12. 12.
    Song, Z.X., Li, J.S., Luo, R.: The Erdös-Jacobson-Lehel conjecture on potentially p k-graphic sequences is true. Science in China, Ser. A 41, 510–520 (1998)MATHMathSciNetGoogle Scholar
  13. 13.
    Justel, C.M., Markenzon, L.: Incremental evaluation of computational circuits. In: Proc. of the Second International Colloquium Journes d’Informatique Messine: JIM 2000 (2000)Google Scholar
  14. 14.
    Kloks, T.: Treewidth, Universiteit Utrecht (1993)Google Scholar
  15. 15.
    Jacobson, M.S., Erdös, P., Lehel, J.: Graphs realizing the degree sequences and their respective clique numbers. In: Alavi, Y., et al. (eds.) Graph Theory, Combinatorics and Applications, vol. 1, pp. 439–449 (1991)Google Scholar
  16. 16.
    Turán, P.: On an extremal problem in graph theory. Mat. Fiz. Lapok 48, 436–452 (1941)MATHMathSciNetGoogle Scholar
  17. 17.
    Uspensky, Y.V.: Asymptotic expressions of numerical functions occurring in problems concerning the partition of numbers into summands. Bull. Acad. Sci. de Russie 14(6), 199–218 (1920)Google Scholar
  18. 18.
    Winkler, P.M.: Graphic Characterization of k-Trees. Congressus Numeratium 33, 349–357 (1981)MathSciNetGoogle Scholar

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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