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The coloring problem for classes with two small obstructions

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Abstract

The coloring problem is studied in the paper for graph classes defined by two small forbidden induced subgraphs. We prove some sufficient conditions for effective solvability of the problem in such classes. As their corollary we determine the computational complexity for all sets of two connected forbidden induced subgraphs with at most five vertices except 13 explicitly enumerated cases.

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References

  1. Kral’, D., Kratochvil, J., Tuza, Z., Woeginger, G.: Complexity of coloring graphs without forbidden induced subgraphs. Lecture Notes in Computer Science, vol. 2204, pp. 254–262 (2001)

  2. Brandstädt, A., Dragan, F., Le, H., Mosca, R.: New graph classes of bounded clique-width. Lecture Notes in Computer Science, vol. 2573, pp. 57–67 (2002)

  3. Dabrowski, K., Lozin, V., Raman, R., Ries, B.: Colouring vertices of triangle-free graphs without forests. Discret. Math. 312, 1372–1385 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  4. Golovach, P., Paulusma, D., Song, J.: Coloring graphs without short cycles and long induced paths. Lecture Notes in Computer Science, vol. 6914, pp. 193–204 (2011)

  5. Golovach P, Paulusma D.: List coloring in the absence of two subgraphs. CIAC, pp. 288–299 (2013)

  6. Schindl, D.: Some new hereditary classes where graph coloring remains NP-hard. Discret. Math. 295, 197–202 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. Alekseev, V.: On easy and hard hereditary classes of graphs with respect to the independent set problem. Discret. Appl. Math. 132, 17–26 (2004)

    Article  Google Scholar 

  8. Alekseev, V., Korobitsyn, D., Lozin, V.: Boundary classes of graphs for the dominating set problem. Discret. Math. 285, 1–6 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. Alekseev, V., Boliac, R., Korobitsyn, D., Lozin, V.: NP-hard graph problems and boundary classes of graphs. Theor. Comput. Sci. 389, 219–236 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. Lozin V, Kaminski M.: Coloring edges and vertices of graphs without short or long cycles. Contrib. Discret. Math. 2(1) (2007)

  11. Malyshev D.: On intersection and symmetric difference of families of boundary classes in the problems of colouring and on the chromatic number. Discret. Math. 24(2), 75–78 (2012) (in Russian). English translation in Discret. Math. Appl. 22(5–6), 645–649 (2012)

  12. Korpeilainen, N., Lozin, V., Malyshev, D., Tiskin, A.: Boundary properties of graphs for algorithmic graph problems. Theor. Comput. Sci. 412, 3544–3554 (2011)

    Google Scholar 

  13. Malyshev D.: A study of boundary graph classes for colorability problems. Discret. Anal. Oper. Res. 19(6), 37–48 (2012) (in Russian). English translation in J. Appl. Ind. Math. 7(2), 221–228 (2013)

  14. Kochol, M., Lozin, V., Randerath, B.: The 3-colorability problem on graphs with maximum degree four. SIAM J. Comput. 32, 1128–1139 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  15. Maffray, F., Preissmann, M.: On the NP-completeness of the \(k\)-colorability problem for triangle-free graphs. Discret. Math. 162, 313–317 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  16. Golumbic, M.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980)

    MATH  Google Scholar 

  17. Courcelle, B., Olariu, S.: Upper bounds to the clique width of graphs. Discret. App. Math. 101, 77–144 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  18. Malyshev, D.: Continual sets of boundary classes of graphs for colorability problems. Discret. Anal. Oper. Res. 16(5), 41–51 (2009). in Russian

    MATH  MathSciNet  Google Scholar 

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Acknowledgments

This study was carried out within “The National Research University Higher School of Economics’ Academic Fund Program” in 2013-2014, research grant No. 12-01-0035. The research is partially supported by Russian Foundation for Basic Research, grant No 14-01-00515-a, and by RF President grant MK-1148.2013.1.

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  1. National Research University Higher School of Economics, 25/12 Bolshaja Pecherskaja Ulitsa, Nizhny Novgorod , 603155, Russia

    D. S. Malyshev

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  1. D. S. Malyshev
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Malyshev, D.S. The coloring problem for classes with two small obstructions. Optim Lett 8, 2261–2270 (2014). https://doi.org/10.1007/s11590-014-0733-y

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  • DOI: https://doi.org/10.1007/s11590-014-0733-y

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