On Chromatic Numbers of Two Extensions of Planar Graphs

Abstract

In this paper, the acyclic chromatic and the circular list chromatic numbers of a simple H-minor free graph G is considered, where H ∈{K5,K3,3}. It is proved that the acyclic chromatic number (resp. the circular list chromatic number) of a simple H-minor free graph G where H ∈{K5,K3,3} is at most 5 (resp. at most 8) and we conclude that G is star 20-colorable. These results generalize the same known results on planar graphs. Moreover, some upper bounds for the coloring numbers of H-minor free graphs for H ∈{K5,K3,3,Kr,s} and r ≤ 2 are obtained. These results generalize some known results and give some new results on group choice number, group chromatic number, and the choice number of the mentioned graphs with much shorter proofs.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Albertson, M.O., Chappell, G.G., Kierstead, H.A., Kündgen, A., Ramamurthi, R.: Coloring with no 2-colored P 4s. Electron. J. Combin. 11(1), 13 (2004)

    MATH  Google Scholar 

  2. 2.

    Borodin, O.V.: On acyclic colorings of planar graphs. Discrete Math. 25(3), 211–236 (1979)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Chuang, H., Lai, H.-J., Omidi, G.R., Zakeri, N.: On group choosability of graphs I. Ars Combin. 126, 195–209 (2016)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Chuang, H., Lai, H.-J., Omidi, G.R., Wang, K., Zakeri, N.: On group choosability of graphs II. Graphs Combin. 30(3), 549–563 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Diestel, R.: Graph Theory. 3rd edition. Springer, Berlin (2005)

  6. 6.

    Erdös, P., Hajnal, A.: On the chromatic number of graphs and set-systems. Acta Math. Acad. Sci. Hungar. 17(1-2), 61–99 (1966)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Fertin, G., Raspaud, A., Reed, B.: On star coloring of graphs. In: WG 2001 27th International Workshop on Graph-Theoretic Concepts in Computer Science, Springer Lecture Notes in Computer Science 2204, pp 140–153 (2001)

  8. 8.

    Grünbaum, B.: Acyclic colorings of planar graphs. Israel J. Math. 14, 390–408 (1973)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Havet, F., Kang, R., Müller, T., Sereni, J.-S.: Circular choosability. J. Graph Theory 61(4), 241–270 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Jaeger, F., Linial, N., Payan, C., Tarsi, M.: Group connectivity of graphs—a nonhomogeneous analogue of nowhere-zero flow properties. J. Combin. Theory Ser. B 56(2), 165–182 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Mohar, B.: Choosability for the Circular Chromatic Number. Problem of the month. http://www.fmf.uni-lj.si/mohar/ (2002)

  12. 12.

    Norine, S., Wong, T.-L., Zhu, X.: Circular choosability via combinatorial Nullstellensatz. J. Graph Theory 59(3), 190–204 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Omidi, G.R.: A note on group choosability of graphs with girth at least 4. Graphs Combin. 27(2), 269–273 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Zhu, X.: Circular choosability of graphs. J. Graph Theory 48(3), 210–218 (2005)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgements

The author wishes to thank Dr. G. R. Omidi for the useful comments and suggestions regarding the manuscript. I also would like to thank the anonymous referee for his/her useful comments.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Khosro Tajbakhsh.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Tajbakhsh, K. On Chromatic Numbers of Two Extensions of Planar Graphs. Acta Math Vietnam 44, 493–500 (2019). https://doi.org/10.1007/s40306-018-0252-5

Download citation

Keywords

  • Acyclic coloring
  • Star coloring
  • Circular choosability
  • Coloring number
  • Planar graphs

Mathematics Subject Classification (2010)

  • 05C15