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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.

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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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Keywords

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