Skip to main content

Complexity separating classes for edge-colouring and total-colouring


The class of unichord-free graphs was recently investigated in a series of papers (Machado et al. in Theor. Comput. Sci. 411:1221–1234, 2010; Machado, de Figueiredo in Discrete Appl. Math. 159:1851–1864, 2011; Trotignon, Vušković in J. Graph Theory 63:31–67, 2010) and proved to be useful with respect to the study of the complexity of colouring problems. In particular, several surprising complexity dichotomies could be found in subclasses of unichord-free graphs. We discuss such results based on the concept of “separating class” and we describe the class of bipartite unichord-free as a final missing separating class with respect to edge-colouring and total-colouring problems, by proving that total-colouring bipartite unichord-free graphs is NP-complete.


  1. 1.

    Borodin O, Kostochka A, Woodall D (1997) List edge and list total colourings of multigraphs. J Comb Theory, Ser A 71:184–204

    MathSciNet  Article  Google Scholar 

  2. 2.

    Campos C, Mello C (2008) The total chromatic number of some bipartite graphs. Ars Comb 88:335–347

    Google Scholar 

  3. 3.

    Johnson D (1985) The NP-completeness column: an ongoing guide. J Algorithms 6:434–451

    MathSciNet  Article  Google Scholar 

  4. 4.

    König D (1916) Graphok és alkalmazásuk a determinánsok és a halmazok elméletére. Math Termtud Ertesito 34:104–119

    Google Scholar 

  5. 5.

    Leven D, Galil Z (1983) NP-completeness of finding the chromatic index of regular graphs. J Algorithms 4:35–44

    MathSciNet  Article  Google Scholar 

  6. 6.

    Machado R, de Figueiredo C (2010) Total chromatic number of {square,unichord}-free graphs. Electron Notes Discrete Math 36:671–678

    Article  Google Scholar 

  7. 7.

    Machado R, de Figueiredo C (2011) Total chromatic number of unichord-free graphs. Discrete Appl Math 159:1851–1864

    MathSciNet  Article  Google Scholar 

  8. 8.

    Machado R, de Figueiredo C, Vušković K (2010) Chromatic index of graphs with no cycle with unique chord. Theor Comput Sci 411:1221–1234

    Article  Google Scholar 

  9. 9.

    Machado R, de Figueiredo C, Trotignon N Edge-colouring and total-colouring chordless graphs. Manuscript available at

  10. 10.

    McDiarmid C, Sánchez-Arroyo A (1994) Total colouring regular bipartite graphs is NP-hard. Discrete Math 124:155–162

    MathSciNet  Article  Google Scholar 

  11. 11.

    Trotignon N, Vušković K (2010) A structure theorem for graphs with no cycle with a unique chord and its consequences. J Graph Theory 63:31–67

    MathSciNet  Article  Google Scholar 

  12. 12.

    Yap HP (1996) Total colourings of graphs. Lecture notes in mathematics, vol 1623. Springer, Berlin

    Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Raphael Machado.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Machado, R., de Figueiredo, C. Complexity separating classes for edge-colouring and total-colouring. J Braz Comput Soc 17, 281–285 (2011).

Download citation


  • Theoretical computer science
  • Computational complexity
  • Colouring of graphs
  • Total chromatic number
  • Bipartite unichord-free