Clique-Colouring and biclique-colouring unichord-free graphs

  • Authors:

  • Hélio B. Macêdo Filho;Raphael C. S. Machado;Celina M. H. Figueiredo

  • Affiliations:

  • COPPE, Universidade Federal do Rio de Janeiro, Brazil;Inmetro -- Instituto Nacional de Metrologia, Qualidade e Tecnologia, Brazil;COPPE, Universidade Federal do Rio de Janeiro, Brazil

  • Venue:
  • LATIN'12 Proceedings of the 10th Latin American international conference on Theoretical Informatics
  • Year:
  • 2012

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Abstract

The class of unichord-free graphs was recently investigated in the context of vertex-colouring [J. Graph Theory 63 (2010) 31---67], edge-colouring [Theoret. Comput. Sci. 411 (2010) 1221---1234] and total-colouring [Discrete Appl. Math. 159 (2011) 1851---1864]. Unichord-free graphs proved to have a rich structure that can be used to obtain interesting results with respect to the study of the complexity of colouring problems. In particular, several surprising complexity dichotomies of colouring problems are found in subclasses of unichord-free graphs. In the present work, we investigate clique-colouring and biclique-colouring problems restricted to unichord-free graphs. We show that the clique-chromatic number of a unichord-free graph is at most 3, and that the 2-clique-colourable unichord-free graphs are precisely those that are perfect. We prove that the biclique-chromatic number of a unichord-free graph is at most its clique-number. We describe an O(nm)-time algorithm that returns an optimal clique-colouring, but the complexity to optimal biclique-colour a unichord-free graph is not classified yet. Nevertheless, we describe an O(n2)-time algorithm that returns an optimal biclique-colouring in a subclass of unichord-free graphs called cactus.