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Complexity separating classes for edge-colouring and total-colouring

Abstract

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.

References

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

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  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. Machado R, de Figueiredo C, Trotignon N Edge-colouring and total-colouring chordless graphs. Manuscript available at http://www.liafa.jussieu.fr/~trot/articles/chordless.pdf

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

    Article  MathSciNet  Google Scholar 

  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

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

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Correspondence to Raphael Machado.

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

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Machado, R., de Figueiredo, C. Complexity separating classes for edge-colouring and total-colouring. J Braz Comput Soc 17, 281–285 (2011). https://doi.org/10.1007/s13173-011-0040-8

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