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Restricted coloring problems on Graphs with few P 4 ’s

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  • Cláudia Linhares-Sales
  • Ana Maia
  • Nicolas Martins
  • Rudini Sampaio

Abstract

In this paper, we obtain linear time algorithms to determine the acyclic chromatic number, the star chromatic number, the non repetitive chromatic number and the clique chromatic number of P 4 -tidy graphs and (q, q − 4)-graphs, for every fixed q, which are the graphs such that every set with at most q vertices induces at most q − 4 distinct P 4 ’s. These classes include cographs and P 4 -sparse graphs. We also obtain a linear time algorithm to compute the harmonious chromatic number of connected P 4 -tidy graphs and connected (q, q − 4)-graphs. All these coloring problems are known to be NP-hard for general graphs. These algorithms are fixed parameter tractable on the parameter q(G), which is the minimum q such that G is a (q, q − 4)-graph. We also prove that every connected (q, q − 4)-graph with at least q vertices is 2-clique-colorable and that every acyclic coloring of a cograph is also nonrepetitive, generalizing the main result of Lyons ( 2011 ). Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • Cláudia Linhares-Sales & Ana Maia & Nicolas Martins & Rudini Sampaio, 2014. "Restricted coloring problems on Graphs with few P 4 ’s," Annals of Operations Research, Springer, vol. 217(1), pages 385-397, June.
    • Handle: RePEc:spr:annopr:v:217:y:2014:i:1:p:385-397:10.1007/s10479-014-1537-2
    DOI: 10.1007/s10479-014-1537-2
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    References listed on IDEAS

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    1. Jaroslaw Grytczuk, 2007. "Nonrepetitive Colorings of Graphs—A Survey," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2007, pages 1-10, January.
    2. Flavia Bonomo & Guillermo Durán & Javier Marenco, 2009. "Exploring the complexity boundary between coloring and list-coloring," Annals of Operations Research, Springer, vol. 169(1), pages 3-16, July.
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