A Polynomial-Time Algorithm for Finding Total Colorings of Partial k-Trees

  • Shuji Isobe
  • Xiao Zhou
  • Takao Nishizeki
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1517)


A total coloring of a graph G is a coloring of all elements of G, i.e. vertices and edges, in such a way that no two adjacent or incident elements receive the same color. Many combinatorial problems can be efficiently solved for partial k-trees (graphs of treewidth bounded by a constant k). However, no polynomial-time algorithm has been known for the problem of finding a total coloring of a given partial k-tree with the minimum number of colors. This paper gives such a first polynomial-time algorithm.


Internal Node Linear Time Algorithm Parallel Time Dynamic Programming Table Halin Graph 
These 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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. ACPS93.
    Arnborg, S., Courcelle, B., Proskurowski, A., Seese, D.: An algebraic theory of graph reduction, J. Assoc. Comput. Mach. 40(5), 1134–1164 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  2. AL91.
    Arnborg, S., Lagergren, J.: Easy problems for tree-decomposable graphs. Journal of Algorithms 12(2), 308–340 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  3. BH95.
    Bodlaender, H.L., Hagerup, T.: Parallel algorithms with optimal speedup for bounded treewidth. In: Fülöp, Z., Gecseg, F. (eds.) ICALP 1995. LNCS, vol. 944, pp. 268–279. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  4. Bod90.
    Bodlaender, H.L.: Polynomial algorithms for graph isomorphism and chromatic index on partial k-trees. Journal of Algorithms 11(4), 631–643 (1990)Google Scholar
  5. Bod96.
    Bodlaender, H.L.: A linear time algorithm for finding tree decompositions of small treewidth. SIAM Journal on Computing 25, 1305–1317 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  6. BPT92.
    Borie, R.B., Parker, R.G., Tovey, C.A.: Automatic generation of linear-time algorithms from predicate calculus descriptions of problems on recursively constructed graph families. Algorithmica 7, 555–581 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Cou90.
    Courcelle, B.: The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Information and Computation 85, 12–75 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  8. FW77.
    Fiorini, S., Wilson, R.J.: Edge-Colourings of Graphs. Pitman, London (1977)zbMATHGoogle Scholar
  9. Sán89.
    Sánchez-Arroyo, A.: Determining the total colouring number is NP-hard. Discrete Math. 78, 315–319 (1989)Google Scholar
  10. Sch94.
    Scheffler, P.: A practical linear time algorithm for disjoint paths in graphs with bounded tree-width, Technical Report 396, Dept. Mathematics, Technische Universität Berlin (1994)Google Scholar
  11. TNS82.
    Takamizawa, K., Nishizeki, T., Saito, N.: Linear-time computability of combinatorial problems on series-parallel graphs. J. Assoc. Comput. Mach. 29(3), 623–641 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  12. TP97.
    Telle, J. A., Proskurowski, A.: Algorithms for vertex partitioning problems on partial k-trees. SIAM J. Discrete Math. 10, 529–550 (1997)Google Scholar
  13. Yap96.
    Yap, H.P.: Total Colourings of Graphs. Lecture Notes in Mathematics, vol. 1623. Springer, Berlin (1996)zbMATHGoogle Scholar
  14. ZN95.
    Zhou, X., Nishizeki, T.: Algorithms for finding f-coloring of partial k-trees. In: Staples, J., Katoh, N., Eades, P., Moffat, A. (eds.) ISAAC 1995. LNCS, vol. 1004, pp. 332–341. Springer, Heidelberg (1995)Google Scholar
  15. ZN98.
    Zhou, X., Nishizeki, T.: The edge-disjoint paths problem is NP-complete for partial k-trees. Submitted to a symposiumGoogle Scholar
  16. ZNN96.
    Zhou, X., Nakano, S., Nishizeki, T.: Edge-coloring partial k-trees. Journal of Algorithms 21, 598–617 (1996)Google Scholar
  17. ZSN96.
    Zhou, X., Suzuki, H., Nishizeki, T.: A linear algorithm for edge-coloring series-parallel multigraphs. Journal of Algorithms 20, 174–201 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  18. ZSN97.
    Zhou, X., Suzuki, H., Nishizeki, T.: An NC parallel algorithm for edge- coloring series-parallel multigraphs. Journal of Algorithms 23, 359–374 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  19. ZTN96.
    Zhou, X., Tamura, S., Nishizeki, T.: Finding edge-disjoint paths in partial k-trees. In: Nagamochi, H., Suri, S., Igarashi, Y., Miyano, S., Asano, T. (eds.) ISAAC 1996. LNCS, vol. 1178, pp. 203–212. Springer, Heidelberg (1996)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Shuji Isobe
    • 1
  • Xiao Zhou
    • 1
  • Takao Nishizeki
    • 1
  1. 1.Graduate School of Information SciencesTohoku UniversityJapan

Personalised recommendations