A Linear 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 1741)


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. The total coloring problem is to find a total coloring of a given graph with the minimum number of colors. Many combinatorial problems can be efficiently solved for partial k-trees, i.e., graphs with bounded tree-width. However, no efficient algorithm has been known for the total coloring problem on partial although a polynomial-time algorithm of very high order has been known. In this paper, we give a linear-time algorithm for the total coloring problem on partial k-trees with bounded k.


Linear Time Generalize Coloring Dynamic Programming Algorithm Linear Algorithm Cient Algorithm 
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.
    S. Arnborg, B. Courcelle, A. Proskurowski and D. Seese, An algebraic theory of graph reduction, J. Assoc. Comput. Mach. 40(5), pp. 1134–1164, 1993.zbMATHMathSciNetGoogle Scholar
  2. AL91.
    S. Arnborg and J. Lagergren, Easy problems for tree-decomposable graphs, J. Algorithms, 12(2), pp. 308–340, 1991.zbMATHCrossRefMathSciNetGoogle Scholar
  3. Bod90.
    H.L. Bodlaender, Polynomial algorithms for graph isomorphism and chromatic index on partial k-trees, Journal of Algorithms, 11(4), pp. 631–643, 1990.zbMATHCrossRefMathSciNetGoogle Scholar
  4. BPT92.
    R. B. Borie, R. G. Parker and C. A. Tovey, Automatic generation of linear-time algorithms from predicate calculus descriptions of problems on recursively constructed graph families, Algorithmica, 7, pp. 555–581, 1992.CrossRefMathSciNetzbMATHGoogle Scholar
  5. Cou90.
    B. Courcelle, The monadic second-order logic of grpahs I: Recognizable sets of finite graphs, Inform. Comput., 85, pp. 12–75, 1990.zbMATHCrossRefMathSciNetGoogle Scholar
  6. CM93.
    B. Courcelle and M. Mosbath, Monadic second-order evaluations on tree-decomposable graphs, Theoret. Comput. Sci., 109, pp.49–82, 1993.zbMATHCrossRefMathSciNetGoogle Scholar
  7. IZN99.
    S. Isobe, X. Zhou and T. Nishizeki, A polynomial-time algorithm for finding total colorings of partial k-trees, Int. J. Found. Comput. Sci., 10(2), pp. 171–194, 1999.CrossRefMathSciNetGoogle Scholar
  8. Sán89.
    A. Sánchez-Arroyo. Determining the total colouring number is NP-hard, Discrete Math., 78, pp. 315–319, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  9. ZNN96.
    X. Zhou, S. Nakano and T. Nishizeki, Edge-coloring partial k-trees, J. Algorithms, 21, pp. 598–617, 1996.zbMATHCrossRefMathSciNetGoogle Scholar
  10. ZSN96.
    X. Zhou, H. Suzuki and T. Nishizeki, A linear algorithm for edge-coloring series-parallel multigraphs, J. Algorithm, 20, pp. 174–201, 1996.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Shuji Isobe
    • 1
  • Xiao Zhou
    • 1
  • Takao Nishizeki
    • 1
  1. 1.Graduate School of Information SciencesTohoku UniversityAoba-yama 05Japan

Personalised recommendations