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The b-Continuity of Graphs with Large Girth

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Abstract

A b-coloring of the vertices of a graph is a proper coloring where each color class contains a vertex which is adjacent to each other color class. The b-chromatic number of G is the maximum integer \(b(G)\) for which G has a b-coloring with \(b(G)\) colors. A graph G is b-continuous if G has a b-coloring with k colors, for every integer k in the interval \([\chi (G),b(G)]\). It is known that not all graphs are b-continuous, and that it is NP-complete to decide whether a given graph G is b-continuous even if \(\chi (G)\) and \(b(G)\) are known. Also, there are many results that show that finding b-colorings of graphs with large girth is an easier task. For instance, finding \(b(G)\) can be done in polynomial time when G has girth at least 7; also, regular graphs with girth at least 8 are b-continuous. In this article, we show that if G has girth at least 10, then G is b-continuous.

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Notes

  1. The graph terminology used in this paper follows [6].

References

  1. Alkhateeb, M., Kohl, A.: Upper bounds on the b-chromatic number and results for restricted graph classes. Discussiones Math. Graph Theory 31, 709–735 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  2. Balakrishnan, R., Kavaskar, T.: b-coloring of Kneser graphs. Discrete Appl. Math. 160, 9–14 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  3. Barth, D., Cohen, J., Faik, T.: On the b-continuity property of graphs. Discrete Appl. Math. 155, 1761–1768 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  4. Betancur Velasquez, C.I., Bonomo, F., Koch, I.: On the b-coloring of \(P_4\)-tidy graphs. Discrete Appl. Math. 159, 67–76 (2011)

    MATH  MathSciNet  Google Scholar 

  5. Blidia, M., Maffray, F., Zemir, Z.: On b-colorings in regular graphs. Discrete Appl. Math. 157, 1787–1793 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bondy, A., Murty, U.S.R.: Graph Theory. Spring-Verlag Press (2008)

  7. Bonomo, F., Duran, G., Maffray, F., Marenco, J., Valencia-Pabon, M.: On the b-coloring of cographs and P4-sparse graphs. Graphs Combin. 25(2), 153–167 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cabello, S., Jakovac, M.: On the b-chromatic number of regular graphs. Discrete Appl. Math. 159, 1303–1310 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  9. Campos, V., Lima, C., Martins, N.A., Sampaio, L., Santos, M.C., Silva, A.: The b-chromatic index of graphs. Discrete Math. 338(11), 2072–2079 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  10. Campos, V., Lima, C., Silva, A.: Graphs with girth at least 7 have high b-chromatic number. Eur. J. Combin. 48, 154–164 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  11. El Sahili, A., Kouider, H.: About b-colouring of regular graphs. Utilitas Math. 80, 211–215 (2009)

    MATH  MathSciNet  Google Scholar 

  12. El Sahili, A., Kouider, M., Mortada, M.: On the b-chromatic number of regular bounded graphs. Discrete Appl. Math. 193, 174–179 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  13. Erdős, P.: On the combinatorial problems which I would most like to see solved. Combinatorica 1, 25–42 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  14. Faik, T.: About the b-continuity of graphs. Electron. Notes Discrete Math. 17, 151–156 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  15. Havet, F., Linhares-Sales, C., Sampaio, L.: b-coloring of tight graphs. Discrete Appl. Math. 160(18), 2709–2715 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  16. Irving, R.W., Manlove, D.F.: The b-chromatic number of a graph. Discrete Appl. Math. 91, 127–141 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  17. Javadi, R., Omoomi, B.: On b-coloring of the Kneser graphs. Discrete Math. 309, 4399–4408 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kára, J., Kratochvíl, J., Voigt, M.: b-Continuity. Preprint no. M14/04, Faculty for Mathematics and Natural Science, Technical University Ilmenau (2004) http://www.mathematik.tu-ilmenau.de/voigt/chord.ps

  19. Kratochvíl, J., Tuza, Zs., Voigt, M.: On the b-chromatic number of graphs. WG 2002. Lecture Notes Comput. Sci. 2573, 310–320 (2002)

  20. Lin, W.-H., Chang, G.J.: b-coloring of tight bipartite graphs and the Erdős–Faber–Lovász Conjecture. Discrete App. Math. 161(7–8), 1060–1066 (2013)

    Article  MATH  Google Scholar 

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

  22. Shaebani, S.: On the b-chromatic number of regular graphs without 4-cycle. Discrete Appl. Math. 160, 1610–1614 (2012)

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to Ana Silva.

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All authors are members of the ParGO Research Group— Parallelism, Graphs and Optimization.

This work was partially supported by CNPq and CAPES, Brazil.

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Sales, C.L., Silva, A. The b-Continuity of Graphs with Large Girth. Graphs and Combinatorics 33, 1139–1146 (2017). https://doi.org/10.1007/s00373-017-1828-x

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  • DOI: https://doi.org/10.1007/s00373-017-1828-x

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